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" }}{PARA 0 "" 0 "" {TEXT -1 158 "This can be seen by visualising the surface of the cylindrical surface \"cut\" at right angles to the base and \"opened out\" to for m a rectangle with dimensions " }{XPPEDIT 18 0 "2*Pi*r" "6#*(\"\"#\"\" \"%#PiGF%%\"rGF%" }{TEXT -1 5 " and " }{TEXT 290 1 "h" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 467 149 149 {PLOTDATA 2 "62-%'CURVESG6$7'7$$\"\"!F)F(7$F($\" \"\"F)7$$\"\"#F)F+7$F.F(F'-%'COLOURG6&%$RGBG$\")!\\DP\"!\")F5$\")viobF 7-%)POLYGONSG6%7&F'F*F-F0-%&COLORG6&F4$\"$`*!\"$$\"$L*FCF+-%&STYLEG6#% ,PATCHNOGRIDG-F$6%7$7$$!3/+++++++5!#=F(F'F1-%*LINESTYLEG6#F/-F$6%7$7$F NF+F*F1FQ-F$6%7$F'7$F($!3%**************\\\"FPF1FQ-F$6%7$F07$F.FfnF1FQ -F$6&7$7$$!\"(!\"#$\"\"'!\"\"7$F`o$\"#5Feo7%7$$!+++++!*!#6$\"++++?&*!# 5Ffo7$$!+++++]F]pF^pFF-F?6&F4$\"#NFbo$\"\"&Feo$\"#vFbo-F$6&7$7$F`o$\" \"%Feo7$F`oF(7%7$Fbp$\"*+++![F`pFbq7$F[pFeqFFFdp-F$6&7$7$$\"$2\"Fbo$Fe oFeo7$$\"$+#FboF^r7%7$$\"+++5s>!\"*$!+++++qF]pF_r7$Fdr$!+++++8F`pFFFdp -F$6&7$7$$\"#$*FboF^r7$F(F^r7%7$$\"*+++z#F`pFjrFbs7$FesFgrFFFdp-%%TEXT G6&7$$!\"'FboFhpQ\"h6\"F1-%%FONTG6$%*HELVETICAG\"\"*-Fis6&7$F+F^rQ&2~~ ~rF_tF1F`t-Fis6&FgtQ\"pF_tF1-Fat6$%'SYMBOLGFdt-%*AXESSTYLEG6#%%NONEG-% +AXESLABELSG6%Q!F_tFfu-Fat6#%(DEFAULTG-%%VIEWG6$FiuFiu" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+ 10" "Curve 11" "Curve 12" "Curve 13" }}{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The surface of a circul ar cone with base radius " }{TEXT 292 1 "r" }{TEXT -1 18 " and slant h eight " }{TEXT 293 1 "d" }{TEXT -1 65 " can be \"flattened out\" to fo rm a sector of a circle with radius " }{TEXT 294 1 "d" }{TEXT -1 67 " \+ and where the angle between the two radius lines of the sector is " } {XPPEDIT 18 0 "theta=2*Pi*r/d" "6#/%&thetaG**\"\"#\"\"\"%#PiGF'%\"rGF' %\"dG!\"\"" }{TEXT -1 10 " radians. 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-%%VIEWG6$FccqFccq" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 72 "The area is given by su btracting the surface area of a cone with radius " }{XPPEDIT 18 0 "r[1 ]" "6#&%\"rG6#\"\"\"" }{TEXT -1 18 " and slant height " }{XPPEDIT 18 0 "d[1]" "6#&%\"dG6#\"\"\"" }{TEXT -1 45 " from the surface area of a \+ cone with radius " }{XPPEDIT 18 0 "r[2]" "6#&%\"rG6#\"\"#" }{TEXT -1 18 " and slant height " }{XPPEDIT 18 0 "d[2];" "6#&%\"dG6#\"\"#" } {TEXT -1 9 " to give " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "S = Pi*r[2]*d[2]-Pi*r[1]*d[1];" "6#/%\"SG,&*(%#PiG\"\"\"&%\"rG6# \"\"#F(&%\"dG6#F,F(F(*(F'F(&F*6#F(F(&F.6#F(F(!\"\"" }{XPPEDIT 18 0 "`` =Pi*(r[2]*d[2]-r[1]*d[1])" "6#/%!G*&%#PiG\"\"\",&*&&%\"rG6#\"\"#F'&%\" dG6#F-F'F'*&&F+6#F'F'&F/6#F'F'!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 295 1 "d " }{TEXT -1 49 " is the slant height of the truncated cone, then " } {XPPEDIT 18 0 "d[2] = d+d[1];" "6#/&%\"dG6#\"\"#,&F%\"\"\"&F%6#F)F)" } {TEXT -1 6 " and: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S = Pi*(r[2]*d[1]+r[2]*d-r[1]*d[1]);" "6#/%\"SG*&%#PiG\"\"\",(*&&%\"r G6#\"\"#F'&%\"dG6#F'F'F'*&&F+6#F-F'F/F'F'*&&F+6#F'F'&F/6#F'F'!\"\"F'" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 22 "By simila r triangles: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d[2] /d[1]=r[2]/r[1]" "6#/*&&%\"dG6#\"\"#\"\"\"&F&6#F)!\"\"*&&%\"rG6#F(F)&F /6#F)F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "so that: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r[2]*d[1]=r[1]*d[2]" "6#/*&&%\"rG6#\"\"#\"\"\"&%\"dG6#F)F)*&&F&6#F)F)&F+6#F(F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r[2]*d[1]=r[1]*d+r[1]*d[1]" "6#/*&&%\"r G6#\"\"#\"\"\"&%\"dG6#F)F),&*&&F&6#F)F)F+F)F)*&&F&6#F)F)&F+6#F)F)F)" } {TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substitu ting for " }{XPPEDIT 18 0 "r[2]*d[1]" "6#*&&%\"rG6#\"\"#\"\"\"&%\"dG6# F(F(" }{TEXT -1 25 " from (ii) in (i) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S = Pi*(r[1]*d+r[1]*d[1]+r[2]*d-r[1]*d[ 1]);" "6#/%\"SG*&%#PiG\"\"\",**&&%\"rG6#F'F'%\"dGF'F'*&&F+6#F'F'&F-6#F 'F'F'*&&F+6#\"\"#F'F-F'F'*&&F+6#F'F'&F-6#F'F'!\"\"F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi*(r[1]*d+r[2]* d)" "6#/%!G*&%#PiG\"\"\",&*&&%\"rG6#F'F'%\"dGF'F'*&&F+6#\"\"#F'F-F'F'F '" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=Pi*(r[1]+r[2])*d" "6#/%!G*(%#PiG\"\"\",&&%\"rG6#F'F'&F*6#\"\"#F'F'% \"dGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S = 2*Pi*r*d;" "6#/% \"SG**\"\"#\"\"\"%#PiGF'%\"rGF'%\"dGF'" }{TEXT -1 17 " ------- (iii), " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 296 7 "_______" }{TEXT -1 20 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "r = (r[1]+r[2])/2;" "6#/%\"rG*&,&&F$6#\"\"\"F)&F$6#\"\" #F)F)F,!\"\"" }{TEXT -1 8 " is the " }{TEXT 259 11 "mean radius" } {TEXT -1 24 " of the truncated cone. " }}{PARA 0 "" 0 "" {TEXT -1 46 " Note that the formula (iii) also holds in the " }{TEXT 259 13 "limitin g case" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "d[1]->infinity" "6#f*6#&%\" dG6#\"\"\"7\"6$%)operatorG%&arrowG6\"%)infinityGF-F-F-" }{TEXT -1 6 " \+ when " }{XPPEDIT 18 0 "r[1]=r[2]" "6#/&%\"rG6#\"\"\"&F%6#\"\"#" } {TEXT -1 53 " and the truncated cone becomes a cylinder of radius " } {TEXT 297 1 "r" }{TEXT -1 12 " and height " }{TEXT 298 1 "d" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "We could use the term " } {TEXT 259 4 "band" }{TEXT -1 80 " to mean either the surface of a trun cated cone or a the surface of a cylinder. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 61 "The area of a surface obtained by rotatin g a curve about the " }{TEXT 339 1 "x" }{TEXT -1 7 " axis " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Suppose that we wish to calculate the area of the surface obtained when the section o f the curve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 15 " from the point" }{XPPEDIT 18 0 "``(a,f(a));" "6#-%!G6$% \"aG-%\"fG6#F&" }{TEXT -1 13 " to the point" }{XPPEDIT 18 0 "``(b,f(b) );" "6#-%!G6$%\"bG-%\"fG6#F&" }{TEXT -1 22 " is rotated about the " } {TEXT 340 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 33 "S uppose that there is a function " }{XPPEDIT 18 0 "s = s(x);" "6#/%\"sG -F$6#%\"xG" }{TEXT -1 49 " which gives the arc length starting at the \+ point" }{XPPEDIT 18 0 "``(a,f(a));" "6#-%!G6$%\"aG-%\"fG6#F&" }{TEXT -1 19 " as far as a value " }{TEXT 313 1 "x" }{TEXT -1 9 " between " } {TEXT 314 1 "a" }{TEXT -1 5 " and " }{TEXT 315 1 "b" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "S=S(x)" "6#/%\"SG- F$6#%\"xG" }{TEXT -1 70 " be the corresponding function which gives th e surface area as far as " }{TEXT 341 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Consider a (small) change " }{XPPEDIT 18 0 "del ta" "6#%&deltaG" }{TEXT 316 1 "x" }{TEXT -1 17 " in the variable " } {TEXT 317 1 "x" }{TEXT -1 9 " and let " }{XPPEDIT 18 0 "delta" "6#%&de ltaG" }{TEXT 318 1 "y" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "delta" "6#%&de ltaG" }{TEXT 319 1 "s" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "delta" "6#% &deltaG" }{TEXT 343 1 "S" }{TEXT -1 33 " be the corresponding changes \+ in " }{TEXT 320 1 "y" }{TEXT -1 2 ", " }{TEXT 321 1 "s" }{TEXT -1 5 " \+ and " }{TEXT 342 1 "S" }{TEXT -1 15 " respectively. 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bnQ$x=aFf]tFf`nFb^t-F]]t6&7$$\"#DFj^nFi^tQ$x=bFf]tFf`nFb^t-F]]t6&7$Fd^ n$!\"&F^bnFa^tFf`nFb^t-F]]t6&7$$\"#VF^bn$\"#vF^bnQ)(a,f(a))Ff]tFf`nFb^ t-F]]t6&7$$\"$Z#F^bn$\"$v\"F^bnQ)(b,f(b))Ff]tFf`nFb^t-F]]t6&7$$\"++++I ;Fgbn$!#:F^bnFa^tFf`nFb^t-F]]t6&7$$\"$(>F^bn$\"$f\"F^bnQ)y~=~f(x)Ff]tF ][lFb^t-F]]t6&7$F`]t$\"%&H\"Fd]tFe]tFedmFg]t-F]]t6&7$$\"#(*F^bnFi\\tQ \"sFf]tFedmFb^t-F]]t6&7$F_^t$\"$H\"F^bnF`btFedmFb^t-%+AXESLABELSG6%Fa^ tQ!Ff]t-Fh]t6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;F(Fd^n;$!$v\" F^bn$\"$w\"F^bn" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "C urve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26 " "Curve 27" "Curve 28" "Curve 29" "Curve 30" }}{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "If the ch anges are small, and the curve is smooth in the sense that the functio n " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 118 " is differ entiable, then a short section of the curve will differ very little fr om a straight line. The small changes " }{XPPEDIT 18 0 "delta" "6#%&de ltaG" }{TEXT 322 1 "x" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "delta" "6#%&de ltaG" }{TEXT 323 1 "y" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "delta" "6#% &deltaG" }{TEXT 324 1 "s" }{TEXT -1 87 " correspond to the lengths of \+ the three sides of an approximate right-angled triangle. " }}{PARA 256 "" 0 "" {GLPLOT2D 278 168 168 {PLOTDATA 2 "6.-%'CURVESG6$7S7$$!35+ ++++++?!#=$!3(************RC\"F*7$$!3%omm;%)R[p\"F*$!3)32]f>(3^5F*7$$! 3fLLe>;KH9F*$!3:M3*p\\R9%))!#>7$$!3$omm\"4'=28\"F*$!3k:OI!*\\4upF77$$! 3/ommTEO,$)F7$!3hU\"F*$\"3OK*[Y^W)HsF7 7$$\"3^*****\\XI8]\"F*$\"3AlLR(=wG-*F77$$\"3Y*****\\KJX!=F*$\"3AiDV3hA \"3\"F*7$$\"3%)*****\\a@n4#F*$\"3!pRE]pAED\"F*7$$\"3iKL3d#e?O#F*$\"3g* ph:iztS\"F*7$$\"3-mmmr#pvn#F*$\"3uC#*[#*4I!f\"F*7$$\"3Kmmm'[[[%HF*$\"3 oG,@\"yCVu\"F*7$$\"3w)**\\PvddD$F*$\"3lz3[yCTA>F*7$$\"3clmmO^'4`$F*$\" 3OL&)eWC3z?F*7$$\"3I***\\PD6H$QF*$\"31fFr7'G*\\AF*7$$\"3V***\\7ON/7%F* $\"3Nj]'\\k(f6CF*7$$\"3&fmmTgO/U%F*$\"3EFH\"*pZAzDF*7$$\"3[mmT&RJfp%F* $\"3J29=@v?KFF*7$$\"3'GLLeu*3$*\\F*$\"3-d')pt!*>'*GF*7$$\"3uJL3dPv,`F* $\"3')[jTM#=a1$F*7$$\"31***\\ioY/d&F*$\"3kxj!4A$z6KF*7$$\"3;KL$3TU1'eF *$\"3-$Q-Q44*oLF*7$$\"31*******)HWghF*$\"3[e/*R#o\">:%F*7$$\"3&**** **\\_?!QwF*$\"3Zs!3]Pc\"4VF*7$$\"3#4L$3x@%>\"zF*$\"3eq(RRv\"p]WF*7$$\" 3x*****\\*3T6#)F*$\"3)>&3jKqR/YF*7$$\"3_kmT?w=$\\)F*$\"3#=d?F.R![ZF*7$ $\"3-++v)[Dxy)F*$\"3wTPaKy;(*[F*7$$\"33mmm\"4!pv!*F*$\"392o2_?'>/&F*7$ $\"3Y)**\\PMirP*F*$\"3gT7'z)>[#>&F*7$$\"3OMLL`f^n'*F*$\"3_1cGV&>kL&F*7 $$\"3GKL$eXWW'**F*$\"3G2!y\")>qD[&F*7$$\"3cm;/C9*e-\"!#<$\"3jA3IedYEcF *7$$\"3++++R,&H0\"Fix$\"3-Hzw]MxddF*7$$\"3smm\"*zC'R3\"Fix$\"3e89xKj=2 fF*7$$\"3ELLL(G+<6\"Fix$\"3VB\">aBX)RgF*7$$\"3****\\PvXFT6Fix$\"3%z1\\ fJq-='F*7$$\"3!***\\iU4ep6Fix$\"3h\\:?,?p8jF*7$$\"3%**************>\"F ix$\"3_***********fX'F*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fb[lFa[l-F$6$ 7&7$Fa[lFa[l7$$\"\"\"Fb[lFa[l7$Fh[l$\"3U+++++++bF*Ff[l-F[[l6&F][lFb[lF b[lFb[l-%)POLYGONSG6$7%Ff[lFg[l7$Fh[l$\"#b!\"#-%&COLORG6&F][l$\"\"'F`[ lFj\\lFh[l-%%TEXTG6&7$$\"#_Ff\\l$!#[!\"$Q\"d6\"F]\\l-%%FONTG6$%'SYMBOL G\"#6-F]]l6&7$$\"$0\"Ff\\l$\"\"$F`[lFe]lF]\\lFg]l-F]]l6&7$$\"#XFf\\l$ \"$U$Fd]lFe]l-F[[l6&F][lFa[lFa[l$\"*++++\"!\")Fg]l-F]]l6&7$$\"#cFf\\l$ !\"&Ff\\lQ\"xFf]lF]\\l-Fh]l6$%*HELVETICAGF_[l-F]]l6&7$$\"$4\"Ff\\lFa^l Q\"yFf]lF]\\lFg_l-F]]l6&7$$\"#\\Ff\\l$\"#MFf\\lQ\"sFf]lFj^lFg_l-%*AXES STYLEG6#%%NONEG-%+AXESLABELSG6%Ff_lQ!Ff]l-Fh]l6#%(DEFAULTG-%%VIEWG6$;$ Ff\\lF`[l$\"#7F`[lFbal" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "This approximation progressively improves as " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 325 1 "x" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 326 1 "y" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 327 1 "s" }{TEXT -1 12 " ten d to 0. " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{XPPEDIT 18 0 "s^2 " "6#*$%\"sG\"\"#" }{TEXT -1 1 " " }{TEXT 328 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "delta" "6#%&deltaG" }{XPPEDIT 18 0 "x^2+delta" "6#,&*$% \"xG\"\"#\"\"\"%&deltaGF'" }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 330 1 "s" }{TEXT -1 1 " " }{TEXT 329 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(delta*x^2+delta*y^2)" "6#-%%sqrtG6#,&*&%&deltaG\"\"\"*$%\"xG\"\" #F)F)*&F(F)*$%\"yGF,F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = delta;" " 6#/%!G%&deltaG" }{XPPEDIT 18 0 "x*sqrt(1+(delta*y/(delta*x))^2);" "6#* &%\"xG\"\"\"-%%sqrtG6#,&F%F%*$*(%&deltaGF%%\"yGF%*&F,F%F$F%!\"\"\"\"#F %F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 95 "The corresponding change in the surface area is approxi mately that of a band with slant height " }{XPPEDIT 18 0 "delta" "6#%& deltaG" }{TEXT 346 1 "s" }{TEXT -1 20 " and average radius " } {XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 101 "Using the formula for the surface area o f a band given in the last section, the corresponding change " } {XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 344 1 "S" }{TEXT -1 39 " in \+ the surface area is approximately: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*y*delta;" "6#** \"\"#\"\"\"%#PiGF%%\"yGF%%&deltaGF%" }{TEXT 406 1 "s" }{TEXT -1 1 " " }{TEXT 405 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*y*delta;" "6#** \"\"#\"\"\"%#PiGF%%\"yGF%%&deltaGF%" }{XPPEDIT 18 0 "x*sqrt(1+(delta*y /(delta*x))^2);" "6#*&%\"xG\"\"\"-%%sqrtG6#,&F%F%*$*(%&deltaGF%%\"yGF% *&F,F%F$F%!\"\"\"\"#F%F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta*S/ (delta*x);" "6#*(%&deltaG\"\"\"%\"SGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 1 " " }{TEXT 404 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*y*sqrt(1+(de lta*y/(delta*x))^2);" "6#**\"\"#\"\"\"%#PiGF%%\"yGF%-%%sqrtG6#,&F%F%*$ *(%&deltaGF%F'F%*&F.F%%\"xGF%!\"\"F$F%F%" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 331 1 "x" }{TEXT -1 13 " tends to 0, " } {XPPEDIT 18 0 "delta*y/(delta*x)" "6#*(%&deltaG\"\"\"%\"yGF%*&F$F%%\"x GF%!\"\"" }{TEXT -1 25 " tends to the derivative " }{XPPEDIT 18 0 "dy/ dx = `f '`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%$f~'G6#%\"xG" }{TEXT -1 8 ". Hence " }{XPPEDIT 18 0 "delta*S/(delta*x);" "6#*(%&deltaG\"\"\"% \"SGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 56 " also tends to a limit which c an only be the derivative " }{XPPEDIT 18 0 "dS/dx = S*`'`(x);" "6#/*&% #dSG\"\"\"%#dxG!\"\"*&%\"SGF&-%\"'G6#%\"xGF&" }{TEXT -1 30 " of the su rface area function " }{XPPEDIT 18 0 "S = S(x);" "6#/%\"SG-F$6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dS/dx = 2*Pi*y*sqrt(1+(dy/dx)^2); " "6#/*&%#dSG\"\"\"%#dxG!\"\"**\"\"#F&%#PiGF&%\"yGF&-%%sqrtG6#,&F&F&*$ *&%#dyGF&F'F(F*F&F&" }{XPPEDIT 18 0 "`` = 2*Pi*f(x)*sqrt(1+`f '`(x)^2) ;" "6#/%!G**\"\"#\"\"\"%#PiGF'-%\"fG6#%\"xGF'-%%sqrtG6#,&F'F'*$-%$f~'G 6#F,F&F'F'" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 20 "It follows that the " }{TEXT 259 12 "surf ace area" }{TEXT -1 62 " of the surface obtained by rotating the secti on of the curve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 22 " from the point where " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\" aG" }{TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x=b" "6#/%\"xG %\"bG" }{TEXT -1 11 " about the " }{TEXT 345 1 "x" }{TEXT -1 32 " axis is given by the integral: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*Pi*Int(y*sqrt(1+(dy/dx)^2),x = a .. b) = 2*Pi*Int(f(x )*sqrt(1+`f '`(x)^2),x = a .. b);" "6#/*(\"\"#\"\"\"%#PiGF&-%$IntG6$*& %\"yGF&-%%sqrtG6#,&F&F&*$*&%#dyGF&%#dxG!\"\"F%F&F&/%\"xG;%\"aG%\"bGF&* (F%F&F'F&-F)6$*&-%\"fG6#F7F&-F.6#,&F&F&*$-%$f~'G6#F7F%F&F&/F7;F9F:F&" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 332 32 "__ ______________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "provided that the derivative " }{XPPEDIT 18 0 "dy/dx = `f '`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%$f~'G6#%\"xG" }{TEXT -1 37 " exi sts throughout the interval from " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\" aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = b" "6#/%\"xG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "A more rigorous derivation of this surface area formula can be made along the following lines. " }} {PARA 0 "" 0 "" {TEXT -1 28 "Subdivide the interval from " }{TEXT 333 1 "a" }{TEXT -1 4 " to " }{TEXT 334 1 "b" }{TEXT -1 6 " into " }{TEXT 335 1 "n" }{TEXT -1 29 " subintervals of equal width " }{XPPEDIT 18 0 "h=(b-a)/n" "6#/%\"hG*&,&%\"bG\"\"\"%\"aG!\"\"F(%\"nGF*" }{TEXT -1 11 " by points " }{XPPEDIT 18 0 "x[0] = a,x[1],` . . . `,x[n] = b;" "6&/& %\"xG6#\"\"!%\"aG&F%6#\"\"\"%(~.~.~.~G/&F%6#%\"nG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "The subinterval are then " } {XPPEDIT 18 0 "[x[i-1],x[i]]" "6#7$&%\"xG6#,&%\"iG\"\"\"F)!\"\"&F%6#F( " }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "i=1,2,` . . . `,n" "6&/%\"iG \"\"\"\"\"#%(~.~.~.~G%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "P[0],P[1],` . . . `,P[n]" "6&&%\"PG6#\"\"! &F$6#\"\"\"%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 28 " be the corresponding p oints" }{XPPEDIT 18 0 "``(x[0],f(x[0])),``(x[1],f(x[1])),` . . . `,``( x[n],f(x[n]));" "6&-%!G6$&%\"xG6#\"\"!-%\"fG6#&F'6#F)-F$6$&F'6#\"\"\"- F+6#&F'6#F3%(~.~.~.~G-F$6$&F'6#%\"nG-F+6#&F'6#F=" }{TEXT -1 17 " along the curve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "For each interval " } {XPPEDIT 18 0 "[x[i-1],x[i]]" "6#7$&%\"xG6#,&%\"iG\"\"\"F)!\"\"&F%6#F( " }{TEXT -1 17 " choose a number " }{XPPEDIT 18 0 "x[i]^`*`" "6#)&%\"x G6#%\"iG%\"*G" }{TEXT -1 71 " in the interval such that the tangent li ne at the corresponding point " }{XPPEDIT 18 0 "P[i]^`*`" "6#)&%\"PG6# %\"iG%\"*G" }{TEXT -1 17 " with coordinates" }{XPPEDIT 18 0 "``(x[i]^` *`,f(x[i]^`*`));" "6#-%!G6$)&%\"xG6#%\"iG%\"*G-%\"fG6#)&F(6#F*F+" } {TEXT -1 52 " is parallel to the line segment joining the points " } {XPPEDIT 18 0 "P[i-1]" "6#&%\"PG6#,&%\"iG\"\"\"F(!\"\"" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "P[i]" "6#&%\"PG6#%\"iG" }{TEXT -1 93 ". It is in tuitively clear that this can be done. More formally, this is a conseq uence of the " }{TEXT 259 18 "mean value theorem" }{TEXT -1 57 " of di fferential calculus. Thus the gradient of the line " }{XPPEDIT 18 0 "P [i-1]*P[i]" "6#*&&%\"PG6#,&%\"iG\"\"\"F)!\"\"F)&F%6#F(F)" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "`f '`(x[i]^`*`);" "6#-%$f~'G6#)&%\"xG6#%\"iG%\" *G" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 339 255 255 {PLOTDATA 2 "6>-%'CURVESG6$7S7$$!35+++++++?!#=$!3B++++++?=F*7$ $!37++](oUIn\"F*$!3)o\"4SR%eg]\"F*7$$!3C+]7y)e&)Q\"F*$!3\"H%o9>z6Q7F*7 $$!35++D\"3F'o5F*$!3\")o#fAM>fU*!#>7$$!3+++](oXdY(F<$!3RviQ5f48lF<7$$! 3!>+]i:F0E%F<$!3yK*fxN/fn$F<7$$!3=-+D\"Gz))G\"F<$!3T3uY=5`+6F<7$$\"3%z **\\(o**3)y\"F<$\"3dAB61AnF<7$$\"34++]7*309\"F*$\"3!3U-Ce(4/ $*F<7$$\"3$****\\i&e*yU\"F*$\"3GAO_!)[a_6F*7$$\"3;++]([D9v\"F*$\"3chqA Fpo'R\"F*7$$\"3!)****\\iNGw?F*$\"38px;DC^N;F*7$$\"39++]7XM*Q#F*$\"35%e U'>Qnf=F*7$$\"3#)**\\(o%QjtEF*$\"3S\"3H9BR\"e?F*7$$\"3'*****\\i8o6IF*$ \"3?/\"[#>C#yG#F*7$$\"3'*******\\>0)H$F*$\"3!f.`vvHqZ#F*7$$\"3z**\\(=- p6j$F*$\"3g55j9?$4p#F*7$$\"3Y*****\\2Mg#RF*$\"3qm\"3ql;Z(GF*7$$\"3A+]( =xZ&\\UF*$\"3Q;.8=fNqIF*7$$\"3G+]i:$4wb%F*$\"3%p4l!QQ\"3D$F*7$$\"3O++v =#R!z[F*$\"3.#[%*\\iKIV$F*7$$\"3#)**\\P4A@u^F*$\"3Q!RR$=i!\\f$F*7$$\"3 I++Dchf#\\&F*$\"3'=$Huc$[Ow$F*7$$\"3L+](of2L#eF*$\"3M(34j6%[KRF*7$$\"3 B+]7yG>6hF*$\"3ixof%f8T2%F*7$$\"3w++voo6AkF*$\"3u+5;'y\"\\@UF*7$$\"3\" ))****\\xJLu'F*$\"3Qf#=fIcwO%F*7$$\"3+++v$*yddqF*$\"3fQq*y.fY]%F*7$$\" 3Q+](=')F*$\"3ka[)=QZy4&F*7$$\"3I,+](Q(zS*)F*$\"3H'>X;/ U:?&F*7$$\"3U**\\(=-,FC*F*$\"3=u1?.-Z$H&F*7$$\"3k+]P4tFe&*F*$\"3-g8Gfd r$Q&F*7$$\"3!)****\\73\"o')*F*$\"37#*\\LA03\"Fiw $\"3a7[9Pnj%o&F*7$$\"38+v=(4bM6\"Fiw$\"3Xv_tl:-XdF*7$$\"37++]xlWU6Fiw$ \"3O4cUhLC&z&F*7$$\"33+]i&3uc<\"Fiw$\"3kf>)41,m%eF*7$$\"3/+++lJR07Fiw$ \"39[Q\"e&Q#p)eF*7$$\"33+v=-*zqB\"Fiw$\"3hZ]S:\"zS#fF*7$$\"33+D\"G:3uE \"Fiw$\"3Yo%f9-**R&fF*7$$\"3/+++++++8Fiw$\"3w************zfF*-%'COLOUR G6&%$RGBG$\"#5!\"\"$\"\"!Fa[lF`[l-F$6$7&7$F`[lF`[l7$$\"\"\"Fa[lF`[l7$F g[l$\"3U+++++++bF*Fe[l-Fjz6&F\\[lFa[lFa[lFa[l-F$6&7%Fe[l7$$\"3++++++++ ]F*$\"3w*************\\$F*Fi[l-%'SYMBOLG6#%'CIRCLEGF\\\\l-%&STYLEG6#%& POINTG-F$6&F`\\l-Fg\\l6#%(DIAMONDGF\\\\lFj\\l-F$6&F`\\l-Fg\\l6#%&CROSS GF\\\\lFj\\l-F$6$7$7$$\"3/+++++++5F*$\"3-+++++++8F*7$$\"3A+++++++!*F*$ \"3]*************p&F*-Fjz6&F\\[lF`[lF`[l$\"*++++\"!\")-F$6%7$7$F`[l$!3 %**************\\\"F*Fe[lF\\\\l-%*LINESTYLEG6#\"\"#-F$6%7$7$Fg[lF^_lFf [lF\\\\lF`_l-F$6%7$7$Fb\\lF^_lFa\\lF\\\\lF`_l-%%TEXTG6&7$$Fi^l!\"#$\" \"&Fa`lQ\"P6\"F\\\\l-%%FONTG6$%*HELVETICAGF^[l-F]`l6&7$$\"#'*Fa`l$\"\" 'F_[lFd`lF\\\\lFf`l-F]`l6&7$$\"#WFa`l$\"#RFa`lFd`lF\\\\lFf`l-F]`l6&7$$ \"%X5!\"$F`[lQ\"QFe`lF\\\\lFf`l-F]`l6&7$$\"#ZFa`l$\"\"%F_[lQ\"*Fe`lF\\ \\lFf`l-F]`l6&7$$F_[lFa`l$Finf inity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "h->0" "6#f*6#%\"hG7\"6$%)operatorG% &arrowG6\"\"\"!F*F*F*" }{TEXT -1 23 ", the three quantities " } {XPPEDIT 18 0 "f(x[i-1]);" "6#-%\"fG6#&%\"xG6#,&%\"iG\"\"\"F+!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "f(x[i]);" "6#-%\"fG6#&%\"xG6#%\"iG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x[i]^`*`);" "6#-%\"fG6#)&%\"xG6#% \"iG%\"*G" }{TEXT -1 34 " approach a common value, so that " } {XPPEDIT 18 0 "``((f(x[i-1])+f(x[i]))/2);" "6#-%!G6#*&,&-%\"fG6#&%\"xG 6#,&%\"iG\"\"\"F0!\"\"F0-F)6#&F,6#F/F0F0\"\"#F1" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "f(x[i]^`*`);" "6#-%\"fG6#)&%\"xG6#%\"iG%\"*G" }{TEXT -1 26 " approach a common value. " }}{PARA 0 "" 0 "" {TEXT -1 97 "This discussion enables us to think of this integral as the limit of a sum in the schematic form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``,delta*x=0) " "6#-%&LimitG6$%!G/*&%&deltaG\"\"\"%\"xGF* \"\"!" }{XPPEDIT 18 0 "Sum(2*Pi*f(x)*sqrt(1+`f '`(x)^2),x = a .. b);" "6#-%$SumG6$**\"\"#\"\"\"%#PiGF(-%\"fG6#%\"xGF(-%%sqrtG6#,&F(F(*$-%$f~ 'G6#F-F'F(F(/F-;%\"aG%\"bG" }{XPPEDIT 18 0 "delta" "6#%&deltaG" } {TEXT 338 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 61 "The area of a surface obtained by rotating a curve abou t the " }{TEXT 288 1 "y" }{TEXT -1 7 " axis " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Suppose that we wish to c alculate the area of the surface obtained when the section of the curv e " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 15 " \+ from the point" }{XPPEDIT 18 0 "``(a,f(a));" "6#-%!G6$%\"aG-%\"fG6#F& " }{TEXT -1 13 " to the point" }{XPPEDIT 18 0 "``(b,f(b));" "6#-%!G6$% \"bG-%\"fG6#F&" }{TEXT -1 22 " is rotated about the " }{TEXT 299 1 "y " }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 33 "Suppose that th ere is a function " }{XPPEDIT 18 0 "s = s(x);" "6#/%\"sG-F$6#%\"xG" } {TEXT -1 49 " which gives the arc length starting at the point" } {XPPEDIT 18 0 "``(a,f(a));" "6#-%!G6$%\"aG-%\"fG6#F&" }{TEXT -1 19 " a s far as a value " }{TEXT 263 1 "x" }{TEXT -1 9 " between " }{TEXT 264 1 "a" }{TEXT -1 5 " and " }{TEXT 265 1 "b" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "S=S(x)" "6#/%\"SG-F$ 6#%\"xG" }{TEXT -1 70 " be the corresponding function which gives the \+ surface area as far as " }{TEXT 300 1 "x" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 26 "Consider a (small) change " }{XPPEDIT 18 0 "delta " "6#%&deltaG" }{TEXT 266 1 "x" }{TEXT -1 17 " in the variable " } {TEXT 267 1 "x" }{TEXT -1 9 " and let " }{XPPEDIT 18 0 "delta" "6#%&de ltaG" }{TEXT 268 1 "y" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "delta" "6#%&de ltaG" }{TEXT 269 1 "s" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "delta" "6#% &deltaG" }{TEXT 302 1 "S" }{TEXT -1 33 " be the corresponding changes \+ in " }{TEXT 270 1 "y" }{TEXT -1 2 ", " }{TEXT 271 1 "s" }{TEXT -1 5 " \+ and " }{TEXT 301 1 "S" }{TEXT -1 15 " respectively. 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__o$!#:Ff^oFfitF`]oFjht-Fbht6&7$$\"$(>Ff^o$\"$f\"Ff^oQ)y~=~f(x)FihtFhz Fjht-Fbht6&7$$\"$d\"Ff^o$\"%&H\"Fb[oQ\"dFihtFijl-F[it6$%'SYMBOLG\"#6-F bht6&7$$\"#(*Ff^oF^htQ\"sFihtFijlFjht-Fbht6&7$$\"$i\"Ff^o$\"$H\"Ff^oFf ]uFijlFjht-%+AXESLABELSG6%FfitQ!Fiht-F[it6#%(DEFAULTG-%*AXESSTYLEG6#%% NONEG-%%VIEWG6$;$!#FF\\[oFd[o;$Ff^oF\\[oFfht" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C urve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "If the changes are small, and the curve i s smooth in the sense that the function " }{XPPEDIT 18 0 "f(x);" "6#-% \"fG6#%\"xG" }{TEXT -1 118 " is differentiable, then a short section o f the curve 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The small ch anges " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 272 1 "x" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 273 1 "y" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 274 1 "s" } {TEXT -1 87 " correspond to the lengths of the three sides of an appro ximate right-angled triangle. 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" }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "del ta" "6#%&deltaG" }{XPPEDIT 18 0 "s^2" "6#*$%\"sG\"\"#" }{TEXT -1 1 " \+ " }{TEXT 278 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "delta" "6#%&deltaG " }{XPPEDIT 18 0 "x^2+delta" "6#,&*$%\"xG\"\"#\"\"\"%&deltaGF'" } {XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 352 1 "s" }{TEXT -1 1 " " }{TEXT 351 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{XPPEDIT 18 0 "x*sqrt(1+(delta*y/(delta*x))^2);" "6#*&%\"xG\"\"\"-%%sqrtG6#,&F% F%*$*(%&deltaGF%%\"yGF%*&F,F%F$F%!\"\"\"\"#F%F%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "The corre sponding change in the surface area is approximately that of a band wi th slant height " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 306 1 "s " }{TEXT -1 20 " and average radius " }{TEXT 353 1 "x" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 100 "Using the formula for the surface ar ea of a band given in the last section the corresponding change " } {XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 303 1 "S" }{TEXT -1 39 " in \+ the surface area is approximately: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*x*delta;" "6#** \"\"#\"\"\"%#PiGF%%\"xGF%%&deltaGF%" }{TEXT 305 1 "s" }{TEXT -1 1 " " }{TEXT 304 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*x*delta;" "6#** \"\"#\"\"\"%#PiGF%%\"xGF%%&deltaGF%" }{XPPEDIT 18 0 "x*sqrt(1+(delta*y /(delta*x))^2);" "6#*&%\"xG\"\"\"-%%sqrtG6#,&F%F%*$*(%&deltaGF%%\"yGF% *&F,F%F$F%!\"\"\"\"#F%F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta*S/ (delta*x);" "6#*(%&deltaG\"\"\"%\"SGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 1 " " }{TEXT 279 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*x*sqrt(1+(de lta*y/(delta*x))^2);" "6#**\"\"#\"\"\"%#PiGF%%\"xGF%-%%sqrtG6#,&F%F%*$ *(%&deltaGF%%\"yGF%*&F.F%F'F%!\"\"F$F%F%" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 280 1 "x" }{TEXT -1 13 " tends to 0, " } {XPPEDIT 18 0 "delta*y/(delta*x)" "6#*(%&deltaG\"\"\"%\"yGF%*&F$F%%\"x GF%!\"\"" }{TEXT -1 25 " tends to the derivative " }{XPPEDIT 18 0 "dy/ dx = `f '`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%$f~'G6#%\"xG" }{TEXT -1 8 ". Hence " }{XPPEDIT 18 0 "delta*S/(delta*x);" "6#*(%&deltaG\"\"\"% \"SGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 56 " also tends to a limit which c an only be the derivative " }{XPPEDIT 18 0 "dS/dx = S*`'`(x);" "6#/*&% #dSG\"\"\"%#dxG!\"\"*&%\"SGF&-%\"'G6#%\"xGF&" }{TEXT -1 30 " of the su rface area function " }{XPPEDIT 18 0 "S = S(x);" "6#/%\"SG-F$6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dS/dx = 2*Pi*x*sqrt(1+(dy/dx)^2); " "6#/*&%#dSG\"\"\"%#dxG!\"\"**\"\"#F&%#PiGF&%\"xGF&-%%sqrtG6#,&F&F&*$ *&%#dyGF&F'F(F*F&F&" }{XPPEDIT 18 0 "`` = 2*Pi*x*sqrt(1+`f '`(x)^2);" "6#/%!G**\"\"#\"\"\"%#PiGF'%\"xGF'-%%sqrtG6#,&F'F'*$-%$f~'G6#F)F&F'F' " }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "It follows that the " }{TEXT 259 12 "surface area" } {TEXT -1 62 " of the surface obtained by rotating the section of the c urve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 22 " from the point where " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" } {TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG " }{TEXT -1 11 " about the " }{TEXT 354 1 "y" }{TEXT -1 32 " axis is g iven by the integral: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*Int(x*sqrt(1+(dy/dx)^2),x = a .. b) = 2*Pi*Int(x*sqrt(1+`f \+ '`(x)^2),x = a .. b);" "6#/*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xGF&-%%sq rtG6#,&F&F&*$*&%#dyGF&%#dxG!\"\"F%F&F&/F,;%\"aG%\"bGF&*(F%F&F'F&-F)6$* &F,F&-F.6#,&F&F&*$-%$f~'G6#F,F%F&F&/F,;F8F9F&" }{TEXT -1 2 ", " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 281 30 "_____________________ _________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "provided tha t the derivative " }{XPPEDIT 18 0 "dy/dx = `f '`(x);" "6#/*&%#dyG\"\" \"%#dxG!\"\"-%$f~'G6#%\"xG" }{TEXT -1 37 " exists throughout the inter val from " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x = b" "6#/%\"xG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "A more rigorous derivation of this surface area formula c an be made along the following lines. " }}{PARA 0 "" 0 "" {TEXT -1 28 "Subdivide the interval from " }{TEXT 282 1 "a" }{TEXT -1 4 " to " } {TEXT 283 1 "b" }{TEXT -1 6 " into " }{TEXT 284 1 "n" }{TEXT -1 29 " s ubintervals of equal width " }{XPPEDIT 18 0 "h=(b-a)/n" "6#/%\"hG*&,&% \"bG\"\"\"%\"aG!\"\"F(%\"nGF*" }{TEXT -1 11 " by points " }{XPPEDIT 18 0 "x[0] = a,x[1],` . . . `,x[n] = b;" "6&/&%\"xG6#\"\"!%\"aG&F%6#\" \"\"%(~.~.~.~G/&F%6#%\"nG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "The subinterval are then " }{XPPEDIT 18 0 "[x[i-1],x[i]] " "6#7$&%\"xG6#,&%\"iG\"\"\"F)!\"\"&F%6#F(" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "i=1,2,` . . . `,n" "6&/%\"iG\"\"\"\"\"#%(~.~.~.~G%\"nG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "P[0],P[1],` . . . `,P[n]" "6&&%\"PG6#\"\"!&F$6#\"\"\"%(~.~.~.~G&F$6#% \"nG" }{TEXT -1 28 " be the corresponding points" }{XPPEDIT 18 0 "``(x [0],f(x[0])),``(x[1],f(x[1])),` . . . `,``(x[n],f(x[n]));" "6&-%!G6$&% \"xG6#\"\"!-%\"fG6#&F'6#F)-F$6$&F'6#\"\"\"-F+6#&F'6#F3%(~.~.~.~G-F$6$& F'6#%\"nG-F+6#&F'6#F=" }{TEXT -1 17 " along the curve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "For each interval " }{XPPEDIT 18 0 "[x[i-1],x[i]]" "6# 7$&%\"xG6#,&%\"iG\"\"\"F)!\"\"&F%6#F(" }{TEXT -1 17 " choose a number \+ " }{XPPEDIT 18 0 "x[i]^`*`" "6#)&%\"xG6#%\"iG%\"*G" }{TEXT -1 71 " in \+ the interval such that the tangent line at the corresponding point " } {XPPEDIT 18 0 "P[i]^`*`" "6#)&%\"PG6#%\"iG%\"*G" }{TEXT -1 17 " with c oordinates" }{XPPEDIT 18 0 "``(x[i]^`*`,f(x[i]^`*`));" "6#-%!G6$)&%\"x G6#%\"iG%\"*G-%\"fG6#)&F(6#F*F+" }{TEXT -1 52 " is parallel to the lin e segment joining the points " }{XPPEDIT 18 0 "P[i-1]" "6#&%\"PG6#,&% \"iG\"\"\"F(!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "P[i]" "6#&%\"PG 6#%\"iG" }{TEXT -1 93 ". It is intuitively clear that this can be done . More formally, this is a consequence of the " }{TEXT 259 18 "mean va lue theorem" }{TEXT -1 57 " of differential calculus. Thus the gradien t of the line " }{XPPEDIT 18 0 "P[i-1]*P[i]" "6#*&&%\"PG6#,&%\"iG\"\" \"F)!\"\"F)&F%6#F(F)" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "`f '`(x[i]^`* `);" "6#-%$f~'G6#)&%\"xG6#%\"iG%\"*G" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 376 272 272 {PLOTDATA 2 "6>-%'CURVESG6$7S7$$!35+++++++?!#=$!3B++++++?=F*7$$!37 ++](oUIn\"F*$!3)o\"4SR%eg]\"F*7$$!3C+]7y)e&)Q\"F*$!3\"H%o9>z6Q7F*7$$!3 5++D\"3F'o5F*$!3\")o#fAM>fU*!#>7$$!3+++](oXdY(F<$!3RviQ5f48lF<7$$!3!>+ ]i:F0E%F<$!3yK*fxN/fn$F<7$$!3=-+D\"Gz))G\"F<$!3T3uY=5`+6F<7$$\"3%z**\\ (o**3)y\"F<$\"3dAB61AnF<7$$\"34++]7*309\"F*$\"3!3U-Ce(4/$*F< 7$$\"3$****\\i&e*yU\"F*$\"3GAO_!)[a_6F*7$$\"3;++]([D9v\"F*$\"3chqAFpo' R\"F*7$$\"3!)****\\iNGw?F*$\"38px;DC^N;F*7$$\"39++]7XM*Q#F*$\"35%eU'>Q nf=F*7$$\"3#)**\\(o%QjtEF*$\"3S\"3H9BR\"e?F*7$$\"3'*****\\i8o6IF*$\"3? /\"[#>C#yG#F*7$$\"3'*******\\>0)H$F*$\"3!f.`vvHqZ#F*7$$\"3z**\\(=-p6j$ F*$\"3g55j9?$4p#F*7$$\"3Y*****\\2Mg#RF*$\"3qm\"3ql;Z(GF*7$$\"3A+](=xZ& \\UF*$\"3Q;.8=fNqIF*7$$\"3G+]i:$4wb%F*$\"3%p4l!QQ\"3D$F*7$$\"3O++v=#R! z[F*$\"3.#[%*\\iKIV$F*7$$\"3#)**\\P4A@u^F*$\"3Q!RR$=i!\\f$F*7$$\"3I++D chf#\\&F*$\"3'=$Huc$[Ow$F*7$$\"3L+](of2L#eF*$\"3M(34j6%[KRF*7$$\"3B+]7 yG>6hF*$\"3ixof%f8T2%F*7$$\"3w++voo6AkF*$\"3u+5;'y\"\\@UF*7$$\"3\"))** **\\xJLu'F*$\"3Qf#=fIcwO%F*7$$\"3+++v$*yddqF*$\"3fQq*y.fY]%F*7$$\"3Q+] (=')F*$\"3ka[)=QZy4&F*7$$\"3I,+](Q(zS*)F*$\"3H'>X;/U:?& F*7$$\"3U**\\(=-,FC*F*$\"3=u1?.-Z$H&F*7$$\"3k+]P4tFe&*F*$\"3-g8Gfdr$Q& F*7$$\"3!)****\\73\"o')*F*$\"37#*\\LA03\"Fiw$\"3 a7[9Pnj%o&F*7$$\"38+v=(4bM6\"Fiw$\"3Xv_tl:-XdF*7$$\"37++]xlWU6Fiw$\"3O 4cUhLC&z&F*7$$\"33+]i&3uc<\"Fiw$\"3kf>)41,m%eF*7$$\"3/+++lJR07Fiw$\"39 [Q\"e&Q#p)eF*7$$\"33+v=-*zqB\"Fiw$\"3hZ]S:\"zS#fF*7$$\"33+D\"G:3uE\"Fi w$\"3Yo%f9-**R&fF*7$$\"3/+++++++8Fiw$\"3w************zfF*-%'COLOURG6&% $RGBG$\"#5!\"\"$\"\"!Fa[lF`[l-F$6$7&7$F`[lF`[l7$$\"\"\"Fa[lF`[l7$Fg[l$ \"3U+++++++bF*Fe[l-Fjz6&F\\[lFa[lFa[lFa[l-F$6&7%Fe[l7$$\"3++++++++]F*$ \"3w*************\\$F*Fi[l-%'SYMBOLG6#%'CIRCLEGF\\\\l-%&STYLEG6#%&POIN TG-F$6&F`\\l-Fg\\l6#%(DIAMONDGF\\\\lFj\\l-F$6&F`\\l-Fg\\l6#%&CROSSGF\\ \\lFj\\l-F$6$7$7$$\"3/+++++++5F*$\"3-+++++++8F*7$$\"3A+++++++!*F*$\"3] *************p&F*-Fjz6&F\\[lF`[lF`[l$\"*++++\"!\")-F$6%7$7$F`[l$!3%*** ***********\\\"F*Fe[lF\\\\l-%*LINESTYLEG6#\"\"#-F$6%7$7$Fg[lF^_lFf[lF \\\\lF`_l-F$6%7$7$Fb\\lF^_lFa\\lF\\\\lF`_l-%%TEXTG6&7$$Fi^l!\"#$\"\"&F a`lQ\"P6\"F\\\\l-%%FONTG6$%*HELVETICAGF^[l-F]`l6&7$$\"#'*Fa`l$\"\"'F_[ lFd`lF\\\\lFf`l-F]`l6&7$$\"#WFa`l$\"#RFa`lFd`lF\\\\lFf`l-F]`l6&7$$\"%X 5!\"$F`[lQ\"QFe`lF\\\\lFf`l-F]`l6&7$$\"#ZFa`l$\"\"%F_[lQ\"*Fe`lF\\\\lF f`l-F]`l6&7$$F_[lFa`l$Finfinity" "6#f*6#%\"nG7\"6$%)operatorG %&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h->0 " "6#f*6#%\"hG7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 23 ", \+ the three quantities " }{XPPEDIT 18 0 "x[i-1];" "6#&%\"xG6#,&%\"iG\"\" \"F(!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[i];" "6#&%\"xG6#%\"iG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "x[i]^`*`;" "6#)&%\"xG6#%\"iG%\"*G" }{TEXT -1 35 " approach a common value, so that " }{XPPEDIT 18 0 "``( (x[i]+x[i-1])/2);" "6#-%!G6#*&,&&%\"xG6#%\"iG\"\"\"&F)6#,&F+F,F,!\"\"F ,F,\"\"#F0" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x[i]^`*`;" "6#)&%\"xG6 #%\"iG%\"*G" }{TEXT -1 26 " approach a common value. " }}{PARA 0 "" 0 "" {TEXT -1 97 "This discussion enables us to think of this integral a s the limit of a sum in the schematic form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``,delta*x=0) " "6#-%&LimitG6$%!G /*&%&deltaG\"\"\"%\"xGF*\"\"!" }{XPPEDIT 18 0 "Sum(2*Pi*x*sqrt(1+`f '` (x)^2),x = a .. b);" "6#-%$SumG6$**\"\"#\"\"\"%#PiGF(%\"xGF(-%%sqrtG6# ,&F(F(*$-%$f~'G6#F*F'F(F(/F*;%\"aG%\"bG" }{XPPEDIT 18 0 "delta" "6#%&d eltaG" }{TEXT 287 1 "x" }{TEXT -1 2 ". 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" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{TEXT 366 16 "________________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exampl e 1 " }}{PARA 0 "" 0 "" {TEXT 355 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 66 "Find the area of the surface formed when \+ the section of the curve " }{XPPEDIT 18 0 "y = sqrt(x);" "6#/%\"yG-%%s qrtG6#%\"xG" }{TEXT -1 26 " from the the point where " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 20 " to the point where " } {XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" }{TEXT -1 22 " is rotated about the " }{TEXT 358 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 356 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 551 "f := x -> sqrt(x):\np1 := plot([f(x),-f(x)],x=0..4.2,y=-2.2..2. 2,color=red,\n thickness=1,linestyle=[1,2]):\na := 1: b := 4:\np2 := plot([f(x),-f(x)],x=a..b,\n color=[COLOR(RGB,.9,0,0),COLOR(RGB,.9, .7,.7)],thickness=3):\np3 := plot([[[a,f(a)],[b,f(b)]]$3],style=point, \n symbol=[circle,diamond,cross],color=black):\nh := evalf(P i/20): r := .08:\np4 := plot([[seq([a+r*f(a)*cos(i*h),f(a)*sin(i*h)],i =0..40)],\n [seq([b+r*f(b)*cos(i*h),f(b)*sin(i*h)],i=0..40)]],\n linestyle=2,color=black):\nplots[display]([p1,p2,p3,p4],tickmar ks=[5,5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 266 300 300 {PLOTDATA 2 "6.-% 'CURVESG6&7W7$$\"\"!F)F(7$$\"3=++](=,()G#!#>$\"3llrW%RXG^\"!#=7$$\"3O+ ++vBSxXF-$\"3Gc3guk[R@F07$$\"3a++]iN5moF-$\"34q!zr3D.i#F07$$\"3s+++]Z! 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" }}{PARA 0 "" 0 "" {TEXT -1 39 "Th e required surface area is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*Int(sqrt(x)*sqrt(1+(dy/dx)^2),x = 1 .. 4) = 2 *Pi*Int(sqrt(x)*sqrt(1+1/(4*x)),x = 1 .. 4);" "6#/*(\"\"#\"\"\"%#PiGF& -%$IntG6$*&-%%sqrtG6#%\"xGF&-F-6#,&F&F&*$*&%#dyGF&%#dxG!\"\"F%F&F&/F/; F&\"\"%F&*(F%F&F'F&-F)6$*&-F-6#F/F&-F-6#,&F&F&*&F&F&*&F:F&F/F&F7F&F&/F /;F&F:F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=Pi*Int(sqrt(4*x+1),x = 1 .. 4)" "6#/%!G*&%#PiG\"\"\" -%$IntG6$-%%sqrtG6#,&*&\"\"%F'%\"xGF'F'F'F'/F1;F'F0F'" }{TEXT -1 8 " \+ ... " }{XPPEDIT 18 0 "PIECEWISE([u=4*x+1,`x =`*1*` implies `*u = 5] ,[du=4*dx,`x =`*4*` implies `*u = 17],[``(1/4)*du=dx,``])" "6#-%*PIEC EWISEG6%7$/%\"uG,&*&\"\"%\"\"\"%\"xGF,F,F,F,/**%%x~~=GF,F,F,%*~implies ~GF,F(F,\"\"&7$/%#duG*&F+F,%#dxGF,/**F0F,F+F,F1F,F(F,\"#<7$/*&-%!G6#*& F,F,F+!\"\"F,F5F,F7F?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi/4" "6#/%!G* &%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(u^(1/2),u= 5..17)" "6#-%$IntG6$)%\"uG*&\"\"\"F)\"\"#!\"\"/F';\"\"&\"#<" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi/4;" "6#/%!G*&%#PiG\"\"\"\"\"%!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``(2/3);" "6#-%!G6#*&\"\"#\"\"\"\"\"$! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u^(3/2)" "6#)%\"uG*&\"\"$\"\"\" \"\"#!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([17, ``],[``, ` `],[1, ``]);" "6#-%*PIECEWISEG6%7$\"#<%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(Pi/6);" "6#/%!G-F$6#*&%#PiG\"\"\"\"\"'! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u^(3/2)" "6#)%\"uG*&\"\"$\"\"\" \"\"#!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([17, ``],[``, ` `],[1, ``]);" "6#-%*PIECEWISEG6%7$\"#<%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi/6" "6#/%!G*&%#PiG\"\"\"\"\"'!\"\"" }{XPPEDIT 18 0 "``(17*sqrt(17)-5*sqrt(5)) " "6#-%!G6#,&*&\"#<\"\"\"-%%sqrtG6#F( F)F)*&\"\"&F)-F+6#F.F)!\"\"" }{TEXT -1 1 " " }{TEXT 357 1 "~" }{TEXT -1 14 " 30.84648970. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "f := x -> sqrt(x):\n'f'(x)=f(x);\nDiff(' f'(x),x)=diff(f(x),x);\n2*Pi*Int(f(x)*sqrt(1+rhs(%)^2),x=1..4);\n``=si mplify(%);\n``=simplify(student[changevar](4*x+1=u,rhs(%),u));\n``=exp and(value(rhs(%)));\n``=evalf(evalf[15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$F'#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,$*&\"\"\"F-*&\"\"#F-F*#F-F/! \"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$Int G6$,$*(F%!\"\"%\"xG#F&F%,&\"\"%F&*&F&F&F.F-F&F/F&/F.;F&F1F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&%#PiG\"\"\"-%$IntG6$*$,&*&\"\"%F '%\"xGF'F'F'F'#F'\"\"#/F/;F'F.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% !G,$*&#\"\"\"\"\"%F(*&%#PiGF(-%$IntG6$*$%\"uG#F(\"\"#/F0;\"\"&\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "a>0" "6#2\"\"!%\"aG" }{TEXT -1 11 " the curve " } {XPPEDIT 18 0 "y = sqrt(a^2-x^2);" "6#/%\"yG-%%sqrtG6#,&*$%\"aG\"\"#\" \"\"*$%\"xGF+!\"\"" }{TEXT -1 59 " is a semi-circle with its centre at the origin and radius " }{TEXT 359 1 "a" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 31 "Rotating this curve around the " }{TEXT 362 1 "x" }{TEXT -1 33 " axis forms a sphere with radius " }{TEXT 361 1 "a" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f( x) = sqrt(a^2-x^2);" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$ F'F.!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence the s urface area of a sphere with radius " }{TEXT 360 1 "a" }{TEXT -1 14 " \+ is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi *Int(f(x)*sqrt(1+`f '`(x)^2),x = -a .. a);" "6#*(\"\"#\"\"\"%#PiGF%-%$ IntG6$*&-%\"fG6#%\"xGF%-%%sqrtG6#,&F%F%*$-%$f~'G6#F.F$F%F%/F.;,$%\"aG! \"\"F:F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "The followin g picture is drawn for the case " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "p1 := plots[tubeplot]([x,0,0],x=-1..1,radius =sqrt(1-x^2),\n tubepoints=30,style=patchnogrid):\np2 := plots[point plot3d]([[-1.8,0,0],[1.8,0,0]],\n style=line,color=brown):\nplots[d isplay]([p1,p2],labels=[`x`,``,`y`],lightmodel=light4,\n scaling=co nstrained,axes=framed,tickmarks=[3,3,3],\n orientation=[-120,60]); " }}{PARA 13 "" 1 "" {GLPLOT3D 390 343 343 {PLOTDATA 3 "6*-%%MESHG6$7T 7@7%$!\"\"\"\"!$F+F+F,F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F( F(F(F(F(F(7@7%$!33xQpMn$=f*!#=$\"3)zntJ0Qy#GF1F,7%F/$\"3-2'[4xC7%F/$\"3m-1(yIwkc#F1$\"3UB]]z$yt=\"F17%F/$\"3[7,2lAA^AF 1$\"3QXW_uYL6F17%F/$!3$4l,*GL0BCF1$\"37\\cy:G!zX\"F 17%F/$!3[Q9o)p4)zEF1$\"3ya5bt,LH!*F97%F/$!3pTaMX6E6GF1$\"3u*35duIu0$F9 7%F/Fap$!3Iq+rX2VdIF97%F/F\\p$!3sh5bt,LH!*F97%F/$!3l];!*GL0BCF1$!3'*\\ cy:G!zX\"F17%F/$!3mtF\\yw*H0#F1$!3zbE*))p-Z%>F17%F/$!3m41QQh%pe\"F1$!3 8`7=X/dSBF17%F/$!31?dG(3P]JF1$\"3oS+5w'\\[R#F17%Fit$\"3i4 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{TEXT -1 6 "Given " }{XPPEDIT 18 0 "f(x) = sqrt(a^2- x^2);" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$F'F.!\"\"" } {XPPEDIT 18 0 "`` = (a^2-x^2)^(1/2);" "6#/%!G),&*$%\"aG\"\"#\"\"\"*$% \"xGF)!\"\"*&F*F*F)F-" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = -1/2;" "6#/-%$f~'G6#%\"xG,$*&\"\"\"F* \"\"#!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "(a^2-x^2)^(-1/2)*`.`*2*x = -x/sqrt(a^2-x^2);" "6#/**),&*$%\"aG\"\"#\"\"\"*$%\"xGF)!\"\",$*&F*F *F)F-F-F*%\".GF*F)F*F,F*,$*&F,F*-%%sqrtG6#,&*$F(F)F**$F,F)F-F-F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+`f '`(x)^2 = 1+x^2/(a^2-x^2);" " 6#/,&\"\"\"F%*$-%$f~'G6#%\"xG\"\"#F%,&F%F%*&F*F+,&*$%\"aGF+F%*$F*F+!\" \"F2F%" }{XPPEDIT 18 0 "`` = (a^2-x^2+x^2)/(a^2-x^2);" "6#/%!G*&,(*$% \"aG\"\"#\"\"\"*$%\"xGF)!\"\"*$F,F)F*F*,&*$F(F)F**$F,F)F-F-" } {XPPEDIT 18 0 "`` = a^2/(a^2-x^2);" "6#/%!G*&%\"aG\"\"#,&*$F&F'\"\"\"* $%\"xGF'!\"\"F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so tha t " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1+`f '`(x) ^2) = a/sqrt(a^2-x^2);" "6#/-%%sqrtG6#,&\"\"\"F(*$-%$f~'G6#%\"xG\"\"#F (*&%\"aGF(-F%6#,&*$F0F.F(*$F-F.!\"\"F6" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)*sqrt(1+`f '`(x)^2) = a;" "6#/*&-%\"fG6#%\"xG\"\"\"-%%sqrtG6 #,&F)F)*$-%$f~'G6#F(\"\"#F)F)%\"aG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 39 "The required surface area is given by: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*Int(f(x)*sqrt(1+`f '`(x)^2) ,x = -a .. a) = 2*Pi*Int(a,x = -a .. a);" "6#/*(\"\"#\"\"\"%#PiGF&-%$I ntG6$*&-%\"fG6#%\"xGF&-%%sqrtG6#,&F&F&*$-%$f~'G6#F/F%F&F&/F/;,$%\"aG! \"\"F;F&*(F%F&F'F&-F)6$F;/F/;,$F;F " 0 "" {MPLTEXT 1 0 161 "a := 'a':\nf := x -> sqrt(a^2-x^2):\n'f'(x)=f(x);\nDiff('f'(x ),x)=diff(f(x),x);\n2*Pi*Int(f(x)*sqrt(1+D(f)(x)^2),x=-a..a);\n``=simp lify(%,symbolic);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"fG6#%\"xG*$,&*$)%\"aG\"\"#\"\"\"F.*$)F'F-F.!\"\"#F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,$*&,&*$)%\"aG\"\" #\"\"\"F2*$)F*F1F2!\"\"#F5F1F*F2F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&,&*$)%\"aGF%F&F&*$)%\"xGF%F&!\"\"#F&F %,&F&F&*&F,F3F2F%F&F4/F2;,$F/F3F/F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\"%#PiGF(%\"aGF(-%$IntG6$F(/%\"xG;,$F*!\"\"F*F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"%\"\"\"%#PiGF()%\"aG\" \"#F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 \+ " }}{PARA 0 "" 0 "" {TEXT 407 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 66 "Find the area of the surface formed when the sect ion of the curve " }{XPPEDIT 18 0 "y = x^2/2;" "6#/%\"yG*&%\"xG\"\"#F' !\"\"" }{TEXT -1 26 " from the the point where " }{XPPEDIT 18 0 "x = 1 ;" "6#/%\"xG\"\"\"" }{TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 22 " is rotated about the " } {TEXT 409 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 408 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 644 "f := x -> x^2/2:\np1 := plot(f(x),x=0..2.4,y=0..2.5,color=red,\n thickne ss=1,linestyle=[1,2]):\np2 := plot(f(-x),x=-2.4..0,y=0..2.5,color=red, \n thickness=1,linestyle=[1,2]):\na := 1: b := 2:\np3 := plot(f(x),x =a..b,color=COLOR(RGB,.9,0,0),thickness=3):\np4 := plot(f(-x),x=-b..-a ,color=COLOR(RGB,.9,.7,.7),thickness=3):\np5 := plot([[[a,f(a)],[b,f(b )]]$3],style=point,\n symbol=[circle,diamond,cross],color=bl ack):\nh := evalf(Pi/20): \np6 := plot([[seq([a*cos(i*h),f(a)+.14*a*si n(i*h)],i=0..40)],\n [seq([b*cos(i*h),f(b)+.14*b*sin(i*h)],i=0.. 40)]],\n linestyle=2,color=black):\nplots[display]([p1,p2,p3,p4, p5,p6],tickmarks=[5,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 408 387 387 {PLOTDATA 2 "6.-%'CURVESG6&7S7$$\"\"!F)F(7$$\"3A********pJJ_!#>$\"3c\\ WsxQLo8!#?7$$\"31+++]z0$y*F-$\"3;5HlU6T&y%F07$$\"3%*******pm>!\\\"!#=$ 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3*4%fvAi.@>F+Fbb_m7%$\"3]R\"*zD7&Qd)F0$!3'>2W(o3!p!=F+Fbb_m7%$\"3$)zD; $\\Hh8\"F+$!3SxINJx'fk\"F+Fbb_m7%$\"3C#p`mq[aQ\"F+$!31(42V*[SU9F+Fbb_m 7%Fdc_m$!3a%)*e\"GX[,7F+Fbb_m7%$\"3a@#R3074x\"F+$!3W?LUQMY%H*F0Fbb_m7% $\"3I4QR$)G2(*=F+$!3Q)eeV()fLL'F0Fbb_m7%Feb_m$!3>D)3jhD#3KF0Fbb_m7%F`b _m$!3EuqH&>(e)*[FewFbb_m-%&STYLEG6#%,PATCHNOGRIDG-%'CURVESG6%7$7%F,F,F ,7%F,F,$\"#E!\"\"-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))eqk\"F`_`mFa_`m-F ^^`m6#%%LINEG-%+AXESLABELSG6%%\"xG%!G%\"yG-%+LIGHTMODELG6#%(LIGHT_2G-% *AXESSTYLEG6#%&FRAMEG-%*AXESTICKSG6%\"\"$Fg``mFg``m-%+PROJECTIONG6%$!$ 5\"F-$\"#qF-\"\"\"" 1 2 0 1 10 0 2 1 4 3 2 1.000000 70.000000 -110.000000 1 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "f(x) = x^2/2;" "6#/-%\"fG6#%\"xG*& F'\"\"#F)!\"\"" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "`f '`(x) = x ;" "6#/-%$f~'G6#%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*sqrt(1+`f '`(x)^2) = x;" "6#/*&%\"xG\"\"\"-%%sqrtG6#,&F&F&*$-%$f~'G6#F%\"\"#F&F &F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1+x^2);" "6#-%%sqrtG6#,&\"\" \"F'*$%\"xG\"\"#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "Th e required surface area is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*Pi*Int(x*sqrt(1+`f '`(x)^2),x = 1 .. 2);" "6#*(\"\"# \"\"\"%#PiGF%-%$IntG6$*&%\"xGF%-%%sqrtG6#,&F%F%*$-%$f~'G6#F+F$F%F%/F+; F%F$F%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*Int(x*sqrt(1+x^2),x = 1 .. 2);" "6#/%!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$*&%\"xGF'-%%sqrtG6#,&F' F'*$F-F&F'F'/F-;F'F&F'" }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWI SE([u = 1+x^2, `x =`*1*` implies `*u = 2],[du = 2*x*dx, `x =`*2*` im plies `*u = 5],[``(1/2)*du = x*dx, ``])" "6#-%*PIECEWISEG6%7$/%\"uG,& \"\"\"F**$%\"xG\"\"#F*/**%%x~~=GF*F*F*%*~implies~GF*F(F*F-7$/%#duG*(F- F*F,F*%#dxGF*/**F0F*F-F*F1F*F(F*\"\"&7$/*&-%!G6#*&F*F*F-!\"\"F*F4F**&F ,F*F6F*F>" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi*Int(u^(1/2),u=2..5)" "6#/%!G*&%#PiG\"\"\"-%$IntG6$)%\"uG*&F'F'\"\"#!\"\"/F,;F.\"\"&F'" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*Pi/3" "6#/%!G*(\"\"#\"\"\"%#PiGF' \"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u^(3/2)" "6#)%\"uG*&\"\"$ \"\"\"\"\"#!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([5,``],[` `,``],[2,``])" "6#-%*PIECEWISEG6%7$\"\"&%!G7$F(F(7$\"\"#F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi/3;" "6#/%!G*(\"\"#\"\"\"%#PiGF'\"\"$!\"\" " }{XPPEDIT 18 0 "``(5*sqrt(5)-2*sqrt(2));" "6#-%!G6#,&*&\"\"&\"\"\"-% %sqrtG6#F(F)F)*&\"\"#F)-F+6#F.F)!\"\"" }{TEXT -1 2 " " }{TEXT 410 1 " ~" }{TEXT -1 14 " 17.49220519. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "2*Pi*Int(x*sqrt(1+x^2),x=1.. 2);\n``=value(%);\n``=evalf(evalf(rhs(%),15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xGF&,&F&F&*$)F,F%F& F&#F&F%/F,;F&F%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\" \"\"%#PiGF(,&*(F'F(\"\"$!\"\"F'#F(F'F-*(\"\"&F(F,F-F0F.F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+>0A\\ " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 "" {TEXT 311 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 66 "Find the area of the surface formed when the section of the curve " } {XPPEDIT 18 0 "y = x^2/2;" "6#/%\"yG*&%\"xG\"\"#F'!\"\"" }{TEXT -1 26 " from the the point where " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG \"\"#" }{TEXT -1 22 " is rotated about the " }{TEXT 363 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 312 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 549 "f := x -> x^2/2:\np1 := plot([f(x),-f(x)],x=0..2.4,y=-2.5..2.5,color=red,\n thickness=1,lin estyle=[1,2]):\na := 1: b := 2:\np2 := plot([f(x),-f(x)],x=a..b,\n \+ color=[COLOR(RGB,.9,0,0),COLOR(RGB,.9,.7,.7)],thickness=3):\np3 := plo t([[[a,f(a)],[b,f(b)]]$3],style=point,\n symbol=[circle,diam ond,cross],color=black):\nh := evalf(Pi/20): r := .08:\np4 := plot([[s eq([a+r*f(a)*cos(i*h),f(a)*sin(i*h)],i=0..40)],\n [seq([b+r*f(b) *cos(i*h),f(b)*sin(i*h)],i=0..40)]],\n linestyle=2,color=black): \nplots[display]([p1,p2,p3,p4],tickmarks=[3,5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 281 460 460 {PLOTDATA 2 "6.-%'CURVESG6&7S7$$\"\"!F)F(7$$\"3A ********pJJ_!#>$\"3c\\WsxQLo8!#?7$$\"31+++]z0$y*F-$\"3;5HlU6T&y%F07$$ \"3%*******pm>!\\\"!#=$\"3XW&Rw0V.6\"F-7$$\"3&********o![0?F9$\"3!zV\" )*)Rw4,#F-7$$\"3l*****\\l:$=DF9$\"3oQ!4\"po&4<$F-7$$\"3q*****\\JzP*HF9 $\"3sL4Y$Hd8[%F-7$$\"3_*****\\R%4'[$F9$\"3Y1_UlqUwgF-7$$\"3]*****\\NZ_ *RF9$\"3=Ys\"Qr+5)zF-7$$\"3k*****\\XnF]%F9$\"3se.pPdu85F97$$\"34+++gU \"[-&F9$\"3_n\\P7&H-(F9$\"3Eu\\#>>#4mCF97$$\"3!))****\\:9yZ(F9$\"3WJ)oZ)F9$\"35Yq ]rt(Gf$F97$$\"3=*****\\Vq)4!*F9$\"3)QNuFE)))eSF97$$\"3)z*****>Xl\"[*F9 $\"3y>=$=i)3&\\%F97$$\"3i*****\\Vw#****F9$\"3%\\Jx6Yw#**\\F97$$\"3-++] !\\<#\\5!#<$\"3F)eb=r'G/bF97$$\"3!******\\FY15\"F^r$\"3owoN866dgF97$$ \"3+++]`R(y9\"F^r$\"3S+'=c1t!)e'F97$$\"3%)*****\\Q:))>\"F^r$\"3'))[jlj \"z&=(F97$$\"3%*****\\:#Hd5/g\"F^r$\"3/$o%[**pl!G\"F^r7$$\"3#)*******3KAl \"F^r$\"3m$)GhRa$\\O\"F^r7$$\"3m****\\(3!>*p\"F^r$\"3FG\"HnZBOW\"F^r7$ $\"3#******>eF0v\"F^r$\"35$QrwSt@`\"F^r7$$\"3u****\\j@$))z\"F^r$\"3%\\ /@id)*yh\"F^r7$$\"3!*****\\pVK\\=F^r$\"3/'G8=J++r\"F^r7$$\"3i******H(* o)*=F^r$\"3\"et.aM6D!=F^r7$$\"3w****\\(oq.&>F^r$\"3%4hH$4H(>!>F^r7$$\" 3(******>fX,+#F^r$\"3M:&)f%>\"H+?F^r7$$\"3%******4iZ50#F^r$\"3az[!=<)R .@F^r7$$\"30++]b\"G:5#F^r$\"3,;'zTH5#3AF^r7$$\"3%******R_9z9#F^r$\"34t ]?,%onI#F^r7$$\"3*)*****p`y5?#F^r$\"3,I5-jLPACF^r7$$\"3u*****R1H'[AF^r $\"3ddFtLj;GDF^r7$$\"3'*****\\VyK*H#F^r$\"3.!=ZfEaMk#F^r7$$\"3))****\\ WI&yM#F^r$\"35fRGfp?cFF^r7$$\"3!**************R#F^r$\"3))************z GF^r-%*LINESTYLEG6#\"\"\"-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKN ESSGFjz-F$6&7SF'7$F+$!3c\\WsxQLo8F07$F2$!3;5HlU6T&y%F07$F7$!3XW&Rw0V.6 \"F-7$F=$!3!zV\")*)Rw4,#F-7$FB$!3oQ!4\"po&4<$F-7$FG$!3sL4Y$Hd8[%F-7$FL $!3Y1_UlqUwgF-7$FQ$!3=Ys\"Qr+5)zF-7$FV$!3se.pPdu85F97$Fen$!3_n\\P>#4mCF97$F^p$!3W=$=i)3&\\%F97$Fgq$!3%\\Jx6Yw#**\\F 97$F\\r$!3F)eb=r'G/bF97$Fbr$!3owoN866dgF97$Fgr$!3S+'=c1t!)e'F97$F\\s$! 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" }}{PARA 0 "" 0 "" {TEXT -1 30 "Th e surface area is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*Pi*Int(f(x)*sqrt(1+`f '`(x)^2),x = 1 .. 2) = Pi*Int(x ^2*sqrt(1+x^2),x = 1 .. 2);" "6#/*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&-%\"fG 6#%\"xGF&-%%sqrtG6#,&F&F&*$-%$f~'G6#F/F%F&F&/F/;F&F%F&*&F'F&-F)6$*&F/F %-F16#,&F&F&*$F/F%F&F&/F/;F&F%F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(x^2*sqrt(1+x^2),x)=1/8" "6#/-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,&\"\"\" F.*$F(F)F.F.F(*&F.F.\"\")!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*(1+2 *x^2)*sqrt(1+x^2)-1/8;" "6#,&*(%\"xG\"\"\",&F&F&*&\"\"#F&*$F%F)F&F&F&- %%sqrtG6#,&F&F&*$F%F)F&F&F&*&F&F&\"\")!\"\"F2" }{TEXT -1 1 " " } {XPPEDIT 18 0 "ln(x+sqrt(1+x^2))+c" "6#,&-%#lnG6#,&%\"xG\"\"\"-%%sqrtG 6#,&F)F)*$F(\"\"#F)F)F)%\"cGF)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "as is shown by two differ ent methods in the following two subsections. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = 1/8;" "6#/-%\"fG6#%\"xG*&\"\"\"F)\"\")!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x*(1+2*x^2)*sqrt(1+x^2)-1/8;" "6#,&*(%\"xG\"\"\",&F&F&* &\"\"#F&*$F%F)F&F&F&-%%sqrtG6#,&F&F&*$F%F)F&F&F&*&F&F&\"\")!\"\"F2" } {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x+sqrt(1+x^2));" "6#-%#lnG6#,&%\"xG \"\"\"-%%sqrtG6#,&F(F(*$F'\"\"#F(F(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)=x^2*sqrt(1+x^2)" "6#/-%\"gG6#%\"xG*&F'\"\"#-%%sqrtG6#,&\"\"\"F .*$F'F)F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "In the fo llowing picture the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 13 " is drawn in " }{TEXT 260 3 "red" }{TEXT -1 20 " while the graph of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 45 "The picture is consistent with the fact that " } {XPPEDIT 18 0 "Diff([f(x)],x) = g(x);" "6#/-%%DiffG6$7#-%\"fG6#%\"xGF+ -%\"gG6#F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "f := x -> (x*(1+2*x^2)*sqrt(1+x^2) -ln(x+sqrt(1+x^2)))/8:\n'f(x)'=f(x);\ng := x -> x^2*sqrt(1+x^2):\n'g(x )'=g(x);\nplot([f(x),g(x)],x=-1.7..1.7,color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&**\"\")!\"\"F'\"\"\",&F,F,*&\" \"#F,)F'F/F,F,F,,&F,F,*$F0F,F,#F,F/F,*&#F,F*F,-%#lnG6#,&F'F,*$F1F3F,F, F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&)F'\"\"#\"\"\",& F+F+*$F)F+F+#F+F*" }}{PARA 13 "" 1 "" {GLPLOT2D 331 488 488 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!3%**************p\"!#<$!3;u;#)=1+zEF*7$$!3ALL$e n*)ei\"F*$!3UWdUyn9\"G#F*7$$!3um;/zmSh:F*$!3I$\\12F*7$$!3KLL30))) ))[\"F*$!3!>yB.hW^m\"F*7$$!3QLLeN-*eT\"F*$!3!*G/Pk9Z$R\"F*7$$!3om;a:'Q KM\"F*$!3[+X[XOud6F*7$$!3WL$3PE\")eF\"F*$!3Kh(e/^(4t'*!#=7$$!30+]PFm81 7F*$!3!z&Q)\\]:,'zFK7$$!3OL$3Zi1S8\"F*$!3cQSUu\"o!RkFK7$$!33+](Q%z5i5F *$!3YUGlI]OUCFK7$$!3c*****\\#RUgxFK$ !3l1&p\"Q(\\X\"=FK7$$!35*****\\5>30(FK$!3k<\"G'RkgH8FK7$$!38ML3Z*HkS'F K$!3G79Q))p.v(*!#>7$$!3\"fmm;\"*)=ScFK$!3$)3y5Q=iAlFcp7$$!35nmmYA3\"* \\FK$!3nA%zg\"QaTWFcp7$$!32++v$)o,OUFK$!37-%p[K\")fm#Fcp7$$!31nmm'4cwc $FK$!3)4tG%eM?q:Fcp7$$!3r***\\PQeV$GFK$!3Sb=2s=Sqx!#?7$$!3))***\\7b&3O @FK$!3Aadn(3/IH$F]r7$$!3Zlm;/6^29FK$!3%4U_#>Bs\\$*!#@7$$!3!GlmTDBXQ(Fc p$!32t!f%yI[W8Fhr7$$!3e`ML$e/#y;F]r$!3EnNzfL[v:!#E7$$\"3dcm;HbIGtFcp$ \"3-ViQmu(RJ\"Fhr7$$\"3E++vB0P&Q\"FK$\"3gm/&RorP\"*)Fhr7$$\"3kmm;p:8!4 #FK$\"3A?r7m(oK3$F]r7$$\"3O)******o=#=GFK$\"35;V\"\\))Gkj(F]r7$$\"3.++ ]#*)40`$FK$\"3(fsEK9a0_\"Fcp7$$\"3@)**\\iD)o>UFK$\"34]\"oT\"HHMEFcp7$$ \"3)4++vK.\\)\\FK$\"3#[j/]IkVU%Fcp7$$\"3GKLL.J[scFK$\"3l\"QN7?G8k'Fcp7 $$\"3c*****\\F@mS'FK$\"3sP()\\)fpfx*Fcp7$$\"3clm\"Hdf=2(FK$\"3!3Ho)R( \\CM\"FK7$$\"3'Q++]uS\"*z(FK$\"3QBp/h&RU%=FK7$$\"35HLe\\cX$[)FK$\"3PI \\o$Rw(HCFK7$$\"3#))**\\7!>w)>*FK$\"3oS\\J[o*o<$FK7$$\"3mHLL3X5)*)*FK$ \"3oMu0NnIfSFK7$$\"3)***\\iS^-j5F*$\"3gJr(G')ec;&FK7$$\"3/nmm)eRN8\"F* $\"3AIO$\\J#**HkFK7$$\"3]mmT'z]c?\"F*$\"3_U,.Hy/\\zFK7$$\"3kL$ep)[;x7F *$\"3Vh22)o:qq*FK7$$\"3<+++4*yGM\"F*$\"3Aq()z)=dm:\"F*7$$\"3fLL3%f%>=9 F*$\"3\"z;J(em\\,9F*7$$\"3ummmSyb&[\"F*$\"3=ZR%*)zS>l\"F*7$$\"3/+]i66Q d:F*$\"3;jgkG,za>F*7$$\"3%***\\PY^7E;F*$\"3#oX%[%=NBG#F*7$$\"3%******* *******p\"F*$\"3;u;#)=1+zEF*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fd[l Fc[l-F$6$7X7$F($\"3wG$Q['4(**p&F*7$$!3qmm\"z$[%Hm\"F*$\"3/GPU&Q?hO&F*7 $F.$\"3'e5qRcXf/&F*7$$!3)**\\Pu<[Of\"F*$\"3Edk!ol^#yZF*7$F3$\"3ATT_U'y /_%F*7$F8$\"3iH;NqQ!f(RF*7$F=$\"3_DZ(e]l]Z$F*7$FB$\"3XK$[Lfm9-$F*7$FG$ \"37)z;.i)*)QEF*7$FM$\"3iYAW.=GzAF*7$FR$\"3!H^-(HYJW>F*7$FW$\"3)4T5x'R iX;F*7$Ffn$\"3)fRujTUFP\"F*7$F[o$\"3g2'))oT%Qf6F*7$F`o$\"3wGg#G7kOZ*FK 7$Feo$\"3RF$))z(z:BwFK7$Fjo$\"3syRlnk*G3'FK7$F_p$\"3+bUa\\,Cu[FK7$Fep$ \"3w6pl[AG_OFK7$Fjp$\"3cg<5R28%y#FK7$F_q$\"3_@(=*Rft[>FK7$Fdq$\"3'4^pP l%R^8FK7$Fiq$\"3K#R**Q3Z+N)Fcp7$F_r$\"3q\"R`9)))zlYFcp7$Fdr$\"3oRV,3[h +?Fcp7$Fjr$\"33-Ng/k'zY&F]r7$F_s$\"3#HO!\\*euj\"G!#B7$Fes$\"3atD([^2[Q &F]r7$Fjs$\"3H#)pE(e\"eP>Fcp7$F_t$\"3C0Yu\\a0jWFcp7$Fdt$\"3KI\"\\%ygt^ #)Fcp7$Fit$\"3K*G(Hu7&=K\"FK7$F^u$\"3S0syq!4E$>FK7$Fcu$\"3=9\"=^?clx#F K7$Fhu$\"3+zl^9>M*p$FK7$F]v$\"3I`#fYltX([FK7$Fbv$\"3>]=bDWKDhFK7$Fgv$ \"3hwFW1C(Qr(FK7$F\\w$\"3'fnyBQ)zP%*FK7$Faw$\"3aJiYUts\\6F*7$Ffw$\"3MZ tee!*\\y8F*7$F[x$\"3!pq(*3;A#\\;F*7$F`x$\"3[Yh**)QjA%>F*7$Fex$\"3'Q+'H \"Q-pF#F*7$Fjx$\"3%)4rRi$fek#F*7$F_y$\"3\"G]vGGG$>IF*7$Fdy$\"3?c'=)pF< !\\$F*7$Fiy$\"3Q5*41gL?&RF*7$$\"3RLe9w%p9_\"F*$\"3Ns\"RmNKY@%F*7$F^z$ \"3w*>M.i!)*)[%F*7$$\"3)*******GJv\"f\"F*$\"3/ZEab\"GGw%F*7$Fcz$\"3E-L ]5w$z/&F*7$$\"3%**\\(=tD1j;F*$\"3M#fT#4$frO&F*7$FhzFi[l-F][l6&F_[lFc[l Fc[lF`[l-%+AXESLABELSG6$Q\"x6\"Q!Fifl-%%VIEWG6$;$F*!\"\"$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "Int(x ^2*sqrt(1+x^2),x)" "6#-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,&\"\"\"F-*$F'F(F -F-F'" }{TEXT -1 18 " by substituting " }{XPPEDIT 18 0 "x=sinh*u" "6# /%\"xG*&%%sinhG\"\"\"%\"uGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 " Int(x^2*sqrt(1+x^2),x)" "6#-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,&\"\"\"F-*$ F'F(F-F-F'" }{TEXT -1 43 " can be found by means of the substitution \+ " }{XPPEDIT 18 0 "x = sinh*u;" "6#/%\"xG*&%%sinhG\"\"\"%\"uGF'" } {TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "u = arcsinh*x;" "6#/%\"uG*&% (arcsinhG\"\"\"%\"xGF'" }{XPPEDIT 18 0 "``=ln(x+sqrt(1+x^2))" "6#/%!G- %#lnG6#,&%\"xG\"\"\"-%%sqrtG6#,&F*F**$F)\"\"#F*F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(x^2*sqrt(1+x^2),x)" "6#-%$IntG6$*&%\"xG\"\"#-%%sqrt G6#,&\"\"\"F-*$F'F(F-F-F'" }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIEC EWISE([x = sinh*u, ``],[dx = cosh*u*du, ``]);" "6#-%*PIECEWISEG6$7$/% \"xG*&%%sinhG\"\"\"%\"uGF+%!G7$/%#dxG*(%%coshGF+F,F+%#duGF+F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(sinh^2*u*sqrt(1+sinh^2*u)*cosh*u,u);" " 6#/%!G-%$IntG6$*,%%sinhG\"\"#%\"uG\"\"\"-%%sqrtG6#,&F,F,*&F)F*F+F,F,F, %%coshGF,F+F,F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(sinh^2*u*cosh^ 2*u,u)" "6#/%!G-%$IntG6$**%%sinhG\"\"#%\"uG\"\"\"%%coshGF*F+F,F+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/4" "6#/%!G*&\"\"\"F&\"\"%!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sinh^2*2*u,u);" "6#-%$IntG6$*(%%sin hG\"\"#F(\"\"\"%\"uGF)F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/8" "6#/%! G*&\"\"\"F&\"\")!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(cosh*4*u -1),u)" "6#-%$IntG6$-%!G6#,&*(%%coshG\"\"\"\"\"%F,%\"uGF,F,F,!\"\"F." }{XPPEDIT 18 0 "``=1/8" "6#/%!G*&\"\"\"F&\"\")!\"\"" }{XPPEDIT 18 0 "` `(sinh*4*u/4-u)+c" "6#,&-%!G6#,&**%%sinhG\"\"\"\"\"%F*%\"uGF*F+!\"\"F* F,F-F*%\"cGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/32" "6#/%!G*&\" \"\"F&\"#K!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*4*u-1/8" "6#,&*( %%sinhG\"\"\"\"\"%F&%\"uGF&F&*&F&F&\"\")!\"\"F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "u+c" "6#,&%\"uG\"\"\"%\"cGF%" }{XPPEDIT 18 0 "``=1/16" "6#/%!G*&\"\"\"F&\"#;!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*2*u*c osh*2*u-1/8;" "6#,&*.%%sinhG\"\"\"\"\"#F&%\"uGF&%%coshGF&F'F&F(F&F&*&F &F&\"\")!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "u+c" "6#,&%\"uG\"\"\" %\"cGF%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/8" "6#/%!G*&\"\"\"F&\"\")! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*u*cosh*u*(1+2*sinh^2*u)-1/8 ;" "6#,&*,%%sinhG\"\"\"%\"uGF&%%coshGF&F'F&,&F&F&*(\"\"#F&*$F%F+F&F'F& F&F&F&*&F&F&\"\")!\"\"F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u+c" "6#,&% \"uG\"\"\"%\"cGF%" }{XPPEDIT 18 0 "``=1/8" "6#/%!G*&\"\"\"F&\"\")!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sqrt(1+x^2)*(1+2*x^2)-1/8" "6#,&*( %\"xG\"\"\"-%%sqrtG6#,&F&F&*$F%\"\"#F&F&,&F&F&*&F,F&*$F%F,F&F&F&F&*&F& F&\"\")!\"\"F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsinh*x+c" "6#,&*&%( arcsinhG\"\"\"%\"xGF&F&%\"cGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/ 8" "6#/%!G*&\"\"\"F&\"\")!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*(1+2 *x^2)*sqrt(1+x^2)-1/8;" "6#,&*(%\"xG\"\"\",&F&F&*&\"\"#F&*$F%F)F&F&F&- %%sqrtG6#,&F&F&*$F%F)F&F&F&*&F&F&\"\")!\"\"F2" }{TEXT -1 1 " " } {XPPEDIT 18 0 "ln(x+sqrt(1+x^2))+c" "6#,&-%#lnG6#,&%\"xG\"\"\"-%%sqrtG 6#,&F)F)*$F(\"\"#F)F)F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "c := 'c':\n Int(x^2*sqrt(1+x^2),x);\n``=student[changevar](u=arcsinh(x),%,u);\n``= simplify(rhs(%));\n``=combine(rhs(%));\n``=value(rhs(%))+c;\n``=expand (rhs(%));\n``=simplify(eval(subs(u=arcsinh(x),rhs(%))));\n``=convert(r hs(%),ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"#\" \"\",&F*F**$F'F*F*#F*F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$In tG6$*&)-%%sinhG6#%\"uG\"\"#\"\"\",&F/F/*$F)F/F/F/F-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G-%$IntG6$*&)-%%sinhG6#%\"uG\"\"#\"\"\")-%%coshGF, F.F/F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,&#\"\"\"\"\")! \"\"*&#F*F+F*-%%coshG6#,$*&\"\"%F*%\"uGF*F*F*F*F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&\"\")!\"\"%\"uG\"\"\"F(*&#F*\"#KF*-%%sinhG6#,$* &\"\"%F*F)F*F*F*F*%\"cGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**& \"\")!\"\"%\"uG\"\"\"F(*&#F*\"\"%F**&-%%sinhG6#F)F*)-%%coshGF1\"\"$F*F *F**&#F*F'F**&F/F*F3F*F*F(%\"cGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G,**&#\"\"\"\"\")F(-%(arcsinhG6#%\"xGF(!\"\"*(F)F.F-F(,&F(F(*$)F-\" \"#F(F(#F(F3F(*(\"\"%F.F-\"\"$F0F4F(%\"cGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&#\"\"\"\"\")F(-%#lnG6#,&%\"xGF(*$,&F(F(*$)F.\" \"#F(F(#F(F3F(F(!\"\"*(F)F5F.F(F0F4F(*(\"\"%F5F.\"\"$F0F4F(%\"cGF(" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "Int(x^2*sqrt(1+x^2) ,x)" "6#-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,&\"\"\"F-*$F'F(F-F-F'" }{TEXT -1 18 " by substituting " }{XPPEDIT 18 0 "x = tan*u;" "6#/%\"xG*&%$ta nG\"\"\"%\"uGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "Int(x^2*sq rt(1+x^2),x)" "6#-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,&\"\"\"F-*$F'F(F-F-F' " }{TEXT -1 40 " can also be found via the substitution " }{XPPEDIT 18 0 "x=tan*u" "6#/%\"xG*&%$tanG\"\"\"%\"uGF'" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2*sqrt(1+x^2),x )" "6#-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,&\"\"\"F-*$F'F(F-F-F'" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "PIECEWISE([x = tan*u, -Pi/2 < u and u < Pi/2],[dx = sec^2*u*du, ``]);" "6#-%*PIECEWISEG6$7$/%\"xG*&%$tanG\"\" \"%\"uGF+32,$*&%#PiGF+\"\"#!\"\"F3F,2F,*&F1F+F2F37$/%#dxG*(%$secGF2F,F +%#duGF+%!G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " ``=Int(tan^2*u*sqrt(1+tan^2*u)*sec^2*u,u) " "6#/%!G-%$IntG6$*,%$tanG\"\"#%\"uG\"\"\"-%%sqrtG6#,&F,F,*&F)F*F+F,F, F,%$secGF*F+F,F+" }{XPPEDIT 18 0 "``=Int(tan^2*u*sec^3*u,u)" "6#/%!G-% $IntG6$**%$tanG\"\"#%\"uG\"\"\"%$secG\"\"$F+F,F+" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(sin^2*u/(cos ^5*u),u);" "6#/%!G-%$IntG6$*(%$sinG\"\"#%\"uG\"\"\"*&%$cosG\"\"&F+F,! \"\"F+" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([v = sin*u, \+ -1 < v and v < 1],[dv = cos*u, ``]);" "6#-%*PIECEWISEG6$7$/%\"vG*&%$si nG\"\"\"%\"uGF+32,$F+!\"\"F(2F(F+7$/%#dvG*&%$cosGF+F,F+%!G" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(v^2/(1 -v^2)^3,v)" "6#/%!G-%$IntG6$*&%\"vG\"\"#*$,&\"\"\"F-*$F)F*!\"\"\"\"$F/ F)" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 46 "We n ow look for a partial fraction expansion " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v^2/((1-v^2)^3) =A/(1+v)+B/(1+v)^2+C/(1+v)^3+ ` D`/(1-v)+E/(1-v)^2+F/(1-v)^3" "6#/*&%\"vG\"\"#*$,&\"\"\"F)*$F%F&!\" \"\"\"$F+,.*&%\"AGF),&F)F)F%F)F+F)*&%\"BGF)*$,&F)F)F%F)F&F+F)*&%\"CGF) *$,&F)F)F%F)F,F+F)*&%#~DGF),&F)F)F%F+F+F)*&%\"EGF)*$,&F)F)F%F+F&F+F)*& %\"FGF)*$,&F)F)F%F+F,F+F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We need to find the constants \+ " }{TEXT 369 1 "A" }{TEXT -1 2 ", " }{TEXT 370 1 "B" }{TEXT -1 2 ", " }{TEXT 371 1 "C" }{TEXT -1 2 ", " }{TEXT 372 1 "D" }{TEXT -1 2 ", " } {TEXT 373 1 "E" }{TEXT -1 5 " and " }{TEXT 374 1 "F" }{TEXT -1 11 " su ch that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v^2=A*(1+ v)^2*(1-v)^3+B*(1+v)*(1-v)^3+C*(1-v)^3+` D`*(1+v)^3*(1-v)^2+E*(1+v)^3* (1-v)+F*(1+v)^3" "6#/*$%\"vG\"\"#,.*(%\"AG\"\"\"*$,&F*F*F%F*F&F*,&F*F* F%!\"\"\"\"$F**(%\"BGF*,&F*F*F%F*F*,&F*F*F%F.F/F**&%\"CGF**$,&F*F*F%F. F/F*F**(%#~DGF**$,&F*F*F%F*F/F*,&F*F*F%F.F&F**(%\"EGF**$,&F*F*F%F*F/F* ,&F*F*F%F.F*F**&%\"FGF**$,&F*F*F%F*F/F*F*" }{TEXT -1 14 " ------- (ii) " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v^2 = A*(1-v^2)^2*(1-v)+B*(1-v^2)*(1-v)^2+C*(1-v)^3 +` D`*(1-v^2)^2*(1+v)+E*(1+v)^2*(1-v^2)+F*(1+v)^3;" "6#/*$%\"vG\"\"#,. *(%\"AG\"\"\"*$,&F*F**$F%F&!\"\"F&F*,&F*F*F%F.F*F**(%\"BGF*,&F*F**$F%F &F.F*,&F*F*F%F.F&F**&%\"CGF**$,&F*F*F%F.\"\"$F*F**(%#~DGF**$,&F*F**$F% F&F.F&F*,&F*F*F%F*F*F**(%\"EGF**$,&F*F*F%F*F&F*,&F*F**$F%F&F.F*F**&%\" FGF**$,&F*F*F%F*F9F*F*" }{TEXT -1 15 " ------- (ii)' " }}{PARA 0 "" 0 "" {TEXT -1 16 "is an identity. " }}{PARA 0 "" 0 "" {TEXT -1 49 "A col lection of equations involving the unknowns " }{TEXT 376 1 "A" }{TEXT -1 2 ", " }{TEXT 377 1 "B" }{TEXT -1 2 ", " }{TEXT 378 1 "C" }{TEXT -1 2 ", " }{TEXT 379 1 "D" }{TEXT -1 2 ", " }{TEXT 380 1 "E" }{TEXT -1 5 " and " }{TEXT 381 1 "F" }{TEXT -1 81 " can be obtained by the co mbined strategies of substituting particular values of " }{TEXT 382 1 "v" }{TEXT -1 58 " in (ii) or (ii)' and comparing coefficients of powe rs of " }{TEXT 375 1 "v" }{TEXT -1 47 " on the left and right sides of (ii) or (ii)'. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[ v = 1]*` . . . `;" "6#*&7#/%\"vG\"\"\"F'%(~.~.~.~GF'" }{TEXT -1 5 " \+ " }{XPPEDIT 18 0 "1 = 8*F;" "6#/\"\"\"*&\"\")F$%\"FGF$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[v = -1]*` . . \+ . `;" "6#*&7#/%\"vG,$\"\"\"!\"\"F(%(~.~.~.~GF(" }{TEXT -1 5 " " } {XPPEDIT 18 0 "1 = 8*C;" "6#/\"\"\"*&\"\")F$%\"CGF$" }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*v^5 ]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"vG\"\"&F'%(~.~.~ .~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = -A+`D `;" "6#/\"\"!,&%\"A G!\"\"%#D~G\"\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*v^4]*` . . . `;" "6#*&7#*(%-coefficie ntsG\"\"\"%#ofGF'%\"vG\"\"%F'%(~.~.~.~GF'" }{TEXT -1 3 " " } {XPPEDIT 18 0 "0 = A-B+` D`-E;" "6#/\"\"!,*%\"AG\"\"\"%\"BG!\"\"%#~DGF '%\"EGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*v]*` . . . `; " "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"vGF'F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = -A-2*B-3*C+` D`+2*E+3*F;" "6#/\"\"!,.% \"AG!\"\"*&\"\"#\"\"\"%\"BGF*F'*&\"\"$F*%\"CGF*F'%#~DGF**&F)F*%\"EGF*F **&F-F*%\"FGF*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[v = 0]*` . . . `;" " 6#*&7#/%\"vG\"\"!\"\"\"%(~.~.~.~GF(" }{TEXT -1 5 " " }{XPPEDIT 18 0 "0 = A+B+C+` D`+E+F;" "6#/\"\"!,.%\"AG\"\"\"%\"BGF'%\"CGF'%#~DGF'%\" EGF'%\"FGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "The 1st an d 2nd equations give " }{XPPEDIT 18 0 "F=1/8" "6#/%\"FG*&\"\"\"F&\"\") !\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C=1/8" "6#/%\"CG*&\"\"\"F& \"\")!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "Adding the 4th and 6th equations gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "0=2*A+C+2*`D `+F" "6#/\"\"!,**&\"\"#\"\"\"%\"AGF(F(%\"C GF(*&F'F(%#D~GF(F(%\"FGF(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A+` D ` = -1/8;" "6#/,&%\"AG\"\"\"%#D~GF&,$*&F&F&\"\")!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Since" }{XPPEDIT 18 0 "` D` = A; " "6#/%#~DG%\"AG" }{TEXT -1 38 " from the 3rd equation it follows that " }{XPPEDIT 18 0 "` D` = A;" "6#/%#~DG%\"AG" }{XPPEDIT 18 0 "``=-1/16 " "6#/%!G,$*&\"\"\"F'\"#;!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "The 3rd equation now becomes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B+E=-1/8" "6#/,&%\"BG\"\"\"%\"EGF&,$*&F&F&\" \")!\"\"F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 31 "while the \+ 5th equation becomes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-2*B+2*E=0" "6#/,&*&\"\"#\"\"\"%\"BGF'!\"\"*&F&F'%\"EGF'F'\"\"! " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "B=E" "6#/%\"BG%\"EG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }{XPPEDIT 18 0 "B=E" "6#/%\"BG%\"EG" } {XPPEDIT 18 0 "``=-1/16" "6#/%!G,$*&\"\"\"F'\"#;!\"\"F)" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The integral (i) now becomes" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/16" "6#*&\"\"\"F$\"#;!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(``(2/((1+v)^3)-1/(1+v)-1/((1+v)^2)+2/((1-v)^3)-1/(1 -v)-1/((1-v)^2)),v);" "6#-%$IntG6$-%!G6#,.*&\"\"#\"\"\"*$,&F,F,%\"vGF, \"\"$!\"\"F,*&F,F,,&F,F,F/F,F1F1*&F,F,*$,&F,F,F/F,F+F1F1*&F+F,*$,&F,F, F/F1F0F1F,*&F,F,,&F,F,F/F1F1F1*&F,F,*$,&F,F,F/F1F+F1F1F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=1/16" "6#/%!G*&\"\"\"F&\"#;!\"\"" }{XPPEDIT 18 0 "``(-1/((1+v)^2)-ln(1+v)+1/(1+v)+1/((1-v)^2)+ln(1-v)-1/(1-v))+c;" "6#, &-%!G6#,.*&\"\"\"F)*$,&F)F)%\"vGF)\"\"#!\"\"F.-%#lnG6#,&F)F)F,F)F.*&F) F),&F)F)F,F)F.F)*&F)F)*$,&F)F)F,F.F-F.F)-F06#,&F)F)F,F.F)*&F)F),&F)F)F ,F.F.F.F)%\"cGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/16" "6#/%!G*&\" \"\"F&\"#;!\"\"" }{XPPEDIT 18 0 "``(1/((1-v)^2)-1/((1+v)^2)+1/(1+v)-1/ (1-v)+ln((1-v)/(1+v)))+c;" "6#,&-%!G6#,,*&\"\"\"F)*$,&F)F)%\"vG!\"\"\" \"#F-F)*&F)F)*$,&F)F)F,F)F.F-F-*&F)F),&F)F)F,F)F-F)*&F)F),&F)F)F,F-F-F --%#lnG6#*&,&F)F)F,F-F),&F)F)F,F)F-F)F)%\"cGF)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/16" "6#/%!G*&\"\"\"F&\"#;!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(((1+v)^2-(1-v)^2+(1-v^2)*(1-v)-(1-v^2)*(1+v))/(1-v^2 )^2+ln((1-v)/(1+v)))+c" "6#,&-%!G6#,&*&,**$,&\"\"\"F,%\"vGF,\"\"#F,*$, &F,F,F-!\"\"F.F1*&,&F,F,*$F-F.F1F,,&F,F,F-F1F,F,*&,&F,F,*$F-F.F1F,,&F, F,F-F,F,F1F,*$,&F,F,*$F-F.F1F.F1F,-%#lnG6#*&,&F,F,F-F1F,,&F,F,F-F,F1F, F,%\"cGF," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/16" "6#/%!G*&\"\"\"F& \"#;!\"\"" }{XPPEDIT 18 0 "``((4*v+(1-v^2)*(-2*v))/(1-v^2)^2+ln((1-v)/ (1+v)) )+c" "6#,&-%!G6#,&*&,&*&\"\"%\"\"\"%\"vGF,F,*&,&F,F,*$F-\"\"#! \"\"F,,$*&F1F,F-F,F2F,F,F,*$,&F,F,*$F-F1F2F1F2F,-%#lnG6#*&,&F,F,F-F2F, ,&F,F,F-F,F2F,F,%\"cGF," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/16" "6#/%!G *&\"\"\"F&\"#;!\"\"" }{XPPEDIT 18 0 "``(2*v*(1+v^2)/((1-v^2)^2)-ln((1+ v)/(1-v)))+c;" "6#,&-%!G6#,&**\"\"#\"\"\"%\"vGF*,&F*F**$F+F)F*F**$,&F* F**$F+F)!\"\"F)F1F*-%#lnG6#*&,&F*F*F+F*F*,&F*F*F+F1F1F1F*%\"cGF*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " Since " }{XPPEDIT 18 0 "tan*u=x" "6#/*&%$tanG\"\"\"%\"uGF &%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sec*u=sqrt(1+x^2)" "6#/*&%$se cG\"\"\"%\"uGF&-%%sqrtG6#,&F&F&*$%\"xG\"\"#F&" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "cos*u=1/sqrt(1+x^2)" "6#/*&%$cosG\"\"\"%\"uGF&*&F&F&-%% sqrtG6#,&F&F&*$%\"xG\"\"#F&!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " v=sin*u" "6#/%\"vG*&%$sinG\"\"\"%\"uGF'" }{XPPEDIT 18 0 "``=cos*u*tan* u" "6#/%!G**%$cosG\"\"\"%\"uGF'%$tanGF'F(F'" }{XPPEDIT 18 0 "``=x/sqrt (1+x^2)" "6#/%!G*&%\"xG\"\"\"-%%sqrtG6#,&F'F'*$F&\"\"#F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "1+v^2=1+x^2/(1+x^2)" "6#/,&\"\"\"F%*$%\"vG\" \"#F%,&F%F%*&%\"xGF(,&F%F%*$F+F(F%!\"\"F%" }{XPPEDIT 18 0 "``=(1+2*x^2 )/(1+x^2)" "6#/%!G*&,&\"\"\"F'*&\"\"#F'*$%\"xGF)F'F'F',&F'F'*$F+F)F'! \"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "1-v^2=1-x^2/(1+x^2)" "6#/,& \"\"\"F%*$%\"vG\"\"#!\"\",&F%F%*&%\"xGF(,&F%F%*$F,F(F%F)F)" }{XPPEDIT 18 0 "``=1/(1+x^2)" "6#/%!G*&\"\"\"F&,&F&F&*$%\"xG\"\"#F&!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Also " }{XPPEDIT 18 0 "(1+v)/(1-v)=(1+x/sqrt(1+x^2))/(1-x/ sqrt(1+x^2))" "6#/*&,&\"\"\"F&%\"vGF&F&,&F&F&F'!\"\"F)*&,&F&F&*&%\"xGF &-%%sqrtG6#,&F&F&*$F-\"\"#F&F)F&F&,&F&F&*&F-F&-F/6#,&F&F&*$F-F3F&F)F)F )" }{XPPEDIT 18 0 "``=(sqrt(1+x^2)+x)/(sqrt(1+x^2)-x)" "6#/%!G*&,&-%%s qrtG6#,&\"\"\"F+*$%\"xG\"\"#F+F+F-F+F+,&-F(6#,&F+F+*$F-F.F+F+F-!\"\"F4 " }{XPPEDIT 18 0 "``=(sqrt(1+x^2)+x)^2/((sqrt(1+x^2)-x)*(sqrt(1+x^2)+x ))" "6#/%!G*&,&-%%sqrtG6#,&\"\"\"F+*$%\"xG\"\"#F+F+F-F+F.*&,&-F(6#,&F+ F+*$F-F.F+F+F-!\"\"F+,&-F(6#,&F+F+*$F-F.F+F+F-F+F+F5" }{XPPEDIT 18 0 " ``=(sqrt(1+x^2)+x)^2" "6#/%!G*$,&-%%sqrtG6#,&\"\"\"F+*$%\"xG\"\"#F+F+F -F+F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "so that " } {XPPEDIT 18 0 "ln((1+v)/(1-v))=2*ln(x+sqrt(1+x^2))" "6#/-%#lnG6#*&,&\" \"\"F)%\"vGF)F),&F)F)F*!\"\"F,*&\"\"#F)-F%6#,&%\"xGF)-%%sqrtG6#,&F)F)* $F2F.F)F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We may conclude that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2*sqrt(1+x^2),x)=1/8" "6#/-%$IntG 6$*&%\"xG\"\"#-%%sqrtG6#,&\"\"\"F.*$F(F)F.F.F(*&F.F.\"\")!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x*(1+2*x^2)*sqrt(1+x^2)-1/8;" "6#,&*(% \"xG\"\"\",&F&F&*&\"\"#F&*$F%F)F&F&F&-%%sqrtG6#,&F&F&*$F%F)F&F&F&*&F&F &\"\")!\"\"F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x+sqrt(1+x^2))+c" "6 #,&-%#lnG6#,&%\"xG\"\"\"-%%sqrtG6#,&F)F)*$F(\"\"#F)F)F)%\"cGF)" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 11 "as before. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 598 "Int (x^2*sqrt(1+x^2),x);\n``=student[changevar](x=tan(u),%,u);\n``=simplif y(rhs(%),\{1+tan(u)^2=sec(u)^2\}):\n``=simplify(rhs(%),symbolic);\n``= student[changevar](v=sin(u),rhs(%),v);\n``=eval(subs(ln=ln@abs,value(r hs(%))));\n simplify(16*map(normal,[selectremove(has,rhs(%),ln)])): \n``=1/16*(-combine(-op(1,%),ln,integer,symbolic)+op(2,%));\n``=eval(s ubs(v=x/sqrt(1+x^2),rhs(%)));\n``=combine(map(simplify,rhs(%),symbolic ),ln,integer,symbolic);\n[selectremove(has,rhs(%),ln)]:\nsimplify(subs op([1,2,1]=map(rationalize,op([1,2,1],%)),\n 2=numer(op([2],%))/e xpand(denom(op([2],%))),%)):\n``=(op(1,%)+op(2,%));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"#\"\"\",&F*F**$F'F*F*#F*F)F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&)-%$tanG6#%\"uG\"\"#\" \"\"),&F/F/*$F)F/F/#\"\"$F.F/F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G-%$IntG6$*&-%$sinG6#%\"uG\"\"#-%$cosGF+!\"&F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&%\"vG\"\"#,&\"\"\"F,*$)F)F*F,!\"\"!\"$F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,.*&\"\"\"F'*&\"#;F'),&%\"vGF' F'F'\"\"#F'!\"\"F.*&F'F'*&F)F'F+F'F.F'*&F'F'*&F)F'),&F,F'F'F.F-F'F.F'* &#F'F)F'-%#lnG6#-%$absG6#F4F'F'*&#F'F)F'-F86#-F;6#F+F'F.*&F'F'*&F)F'F4 F'F.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"#;F(-%#lnG6 #*&-%$absG6#,&%\"vGF(F(!\"\"F3-F/6#,&F2F(F(F(F(F(F3*,\"\")F3F2F(,&F(F( *$)F2\"\"#F(F(F(F6!\"#F1F=F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,& *&#\"\"\"\"#;F(-%#lnG6#*&-%$absG6#,&*&%\"xGF(,&F(F(*$)F3\"\"#F(F(#!\" \"F7F9F(F(F9-F/6#,&F2F(F(F(F(F(F9*.\"\")F9F3F(F4F8,&F(F(*&F3F7F4F9F(F( F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 29 "The required surface area is " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Pi*Int(x^2*sqrt(1+x^2),x=1..2)" "6#*& %#PiG\"\"\"-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,&F%F%*$F*F+F%F%/F*;F%F+F%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = Pi/8;" "6#/%!G*&%#PiG\"\"\"\"\")!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(x*(1+2*x^2)*sqrt(1+x^2)-ln(x+sqrt(1+x^2)));" "6#-%!G6#,&*(%\"xG \"\"\",&F)F)*&\"\"#F)*$F(F,F)F)F)-%%sqrtG6#,&F)F)*$F(F,F)F)F)-%#lnG6#, &F(F)-F/6#,&F)F)*$F(F,F)F)!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIEC EWISE([2,``],[``,``],[1,``])" "6#-%*PIECEWISEG6%7$\"\"#%!G7$F(F(7$\"\" \"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi/8;" "6#/%!G*&%#PiG\"\"\"\" \")!\"\"" }{XPPEDIT 18 0 "``(``(18*sqrt(5)-ln(2+sqrt(5))-(3*sqrt(2)-ln (1+sqrt(2)))" "6#-%!G6#-F$6#,(*&\"#=\"\"\"-%%sqrtG6#\"\"&F+F+-%#lnG6#, &\"\"#F+-F-6#F/F+!\"\",&*&\"\"$F+-F-6#F4F+F+-F16#,&F+F+-F-6#F4F+F7F7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Pi(9*sqrt(5)/4-3/8*sqrt(2))-Pi/8; " "6#/%!G,&-%#PiG6#,&*(\"\"*\"\"\"-%%sqrtG6#\"\"&F,\"\"%!\"\"F,*(\"\"$ F,\"\")F2-F.6#\"\"#F,F2F,*&F'F,F5F2F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((2+sqrt(5))/(1+sqrt(2)));" "6#-%#lnG6#*&,&\"\"#\"\"\"-%%sqrtG6#\" \"&F)F),&F)F)-F+6#F(F)!\"\"" }{TEXT -1 1 " " }{TEXT 368 1 "~" }{TEXT -1 14 " 13.91895232. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Pi*Int(x^2*sqrt(1+x^2),x=1..2);\n`` =value(%);\n``=evalf(evalf(rhs(%),15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"\"\"-%$IntG6$*&)%\"xG\"\"#F%,&F%F%*$F*F%F%#F%F,/F+;F%F, F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&%#PiG\"\"\",**(\"\"$F'\"\" )!\"\"\"\"##F'F-F,*&#F'F+F'-%#lnG6#,&F'F'*$F-F.F'F'F'*(\"\"*F'\"\"%F, \"\"&F.F'*&F0F'-F26#,&F-F,*$F9F.F'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+K_*=R\"!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{PARA 0 "" 0 "" {TEXT 387 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 66 "Find the area of the su rface formed when the section of the curve " }{XPPEDIT 18 0 "y = exp(x );" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 26 " from the the point where \+ " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 20 " to the point \+ where " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 22 " is rot ated about the " }{TEXT 389 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 388 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 552 "f := x -> exp(x):\np1 := plot([f(x),-f(x)],x=-.2..1. 3,y=-3.2..3.2,color=red,\n thickness=1,linestyle=[1,2]):\na := 0: b \+ := 1:\np2 := plot([f(x),-f(x)],x=a..b,\n color=[COLOR(RGB,.9,0,0),C OLOR(RGB,.9,.7,.7)],thickness=3):\np3 := plot([[[a,f(a)],[b,f(b)]]$3], style=point,\n symbol=[circle,diamond,cross],color=black):\n h := evalf(Pi/20): r := .08:\np4 := plot([[seq([a+r*f(a)*cos(i*h),f(a) *sin(i*h)],i=0..40)],\n [seq([b+r*f(b)*cos(i*h),f(b)*sin(i*h)],i =0..40)]],\n linestyle=2,color=black):\nplots[display]([p1,p2,p3 ,p4],tickmarks=[3,5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 166 358 358 {PLOTDATA 2 "6.-%'CURVESG6&7S7$$!35+++++++?!#=$\"3A=)z2`2t=)F*7$$!37++ ](oUIn\"F*$\"3#R!e0%y@%f%)F*7$$!3C+]7y)e&)Q\"F*$\"3T(y+'zW`.()F*7$$!35 ++D\"3F'o5F*$\"3;_DL$\"3W@`;HLh!G*F*7$$!3! >+]i:F0E%F?$\"3#fg%p&z&*Ge*F*7$$!3=-+D\"Gz))G\"F?$\"3?\"\\Y(>\"R>()*F* 7$$\"3%z**\\(o**3)y\"F?$\"3D0@3?37G;437\"FQ7$$\"3$****\\i&e*yU\"F*$\"3;?$pe1([`6FQ7$$\"3;++]([D9v\"F *$\"3+-_PRgT\">\"FQ7$$\"3!)****\\iNGw?F*$\"3=of)=ob2B\"FQ7$$\"39++]7XM *Q#F*$\"3C'\\1QH&*)p7FQ7$$\"3#)**\\(o%QjtEF*$\"3@[3nF^^18FQ7$$\"3'**** *\\i8o6IF*$\"3GX*4zaO9N\"FQ7$$\"3'*******\\>0)H$F*$\"3E<(3t=(p!R\"FQ7$ $\"3z**\\(=-p6j$F*$\"3?>=%=$R!yV\"FQ7$$\"3Y*****\\2Mg#RF*$\"3k#>%R')4$ 3[\"FQ7$$\"3A+](=xZ&\\UF*$\"3h*oY)[7_H:FQ7$$\"3G+]i:$4wb%F*$\"3sX.a*>t td\"FQ7$$\"3O++v=#R!z[F*$\"3hLd&4M)*)G;FQ7$$\"3#)**\\P4A@u^F*$\"3;)4O+ m&px;FQ7$$\"3I++Dchf#\\&F*$\"3BJcN>-(>t\"FQ7$$\"3L+](of2L#eF*$\"3O!**p @61-z\"FQ7$$\"3B+]7yG>6hF*$\"3s+t\\CD\\U=FQ7$$\"3w++voo6AkF*$\"33K(pH% *z1!>FQ7$$\"3\"))****\\xJLu'F*$\"3eC&\\6vBF'>FQ7$$\"3+++v$*yddqF*$\"3, u_n:4QD?FQ7$$\"3Q+](=')F*$\"3WpJyuq(yO#FQ7$$\"3I,+](Q(zS*)F*$ \"3MAD:PY3XCFQ7$$\"3U**\\(=-,FC*F*$\"3@x<$eAG+_#FQ7$$\"3k+]P4tFe&*F*$ \"3p(oFMZA3g#FQ7$$\"3!)****\\73\"o')*F*$\"3%*)4'o*G3\"FQ$\"3y\\hFW8$FQ7$$\"33+]i&3uc<\"FQ$\"3tKcIlkKSKFQ7$$\"3/+++lJR07 FQ$\"349,)GBr!QLFQ7$$\"33+v=-*zqB\"FQ$\"3'ferSXPbW$FQ7$$\"33+D\"G:3uE \"FQ$\"3og(\\mKN;b$FQ7$$\"3/+++++++8FQ$\"3+W#>wm'HpOFQ-%*LINESTYLEG6# \"\"\"-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fe[lFd[l-%*THICKNESSGF[[l- F$6&7S7$F($!3A=)z2`2t=)F*7$F.$!3#R!e0%y@%f%)F*7$F3$!3T(y+'zW`.()F*7$F8 $!3;_DL\"R>()*F*7$FM$!3D0@3?37G;437\"FQ7$F\\o$!3;?$pe1([`6FQ7$Fao$!3+-_PRgT\">\"FQ7$Ffo$ !3=of)=ob2B\"FQ7$F[p$!3C'\\1QH&*)p7FQ7$F`p$!3@[3nF^^18FQ7$Fep$!3GX*4za O9N\"FQ7$Fjp$!3E<(3t=(p!R\"FQ7$F_q$!3?>=%=$R!yV\"FQ7$Fdq$!3k#>%R')4$3[ \"FQ7$Fiq$!3h*oY)[7_H:FQ7$F^r$!3sX.a*>ttd\"FQ7$Fcr$!3hLd&4M)*)G;FQ7$Fh r$!3;)4O+m&px;FQ7$F]s$!3BJcN>-(>t\"FQ7$Fbs$!3O!**p@61-z\"FQ7$Fgs$!3s+t \\CD\\U=FQ7$F\\t$!33K(pH%*z1!>FQ7$Fat$!3eC&\\6vBF'>FQ7$Fft$!3,u_n:4QD? 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" }}{PARA 0 "" 0 "" {TEXT -1 21 "The surface area is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*Int(f(x)*sqrt(1+`f '`(x)^2),x = 0 .. 1);" "6#*(\"\"#\"\"\"% #PiGF%-%$IntG6$*&-%\"fG6#%\"xGF%-%%sqrtG6#,&F%F%*$-%$f~'G6#F.F$F%F%/F. ;\"\"!F%F%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=2*Pi*Int(exp(x)*sqrt(1+exp(2*x)),x=0..1)" "6#/%!G*( \"\"#\"\"\"%#PiGF'-%$IntG6$*&-%$expG6#%\"xGF'-%%sqrtG6#,&F'F'-F.6#*&F& F'F0F'F'F'/F0;\"\"!F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "The indefinite integral " }{XPPEDIT 18 0 "Int(exp(x)*sqrt(1+exp(2* x)),x)" "6#-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%%sqrtG6#,&F+F+-F(6#*&\"\"# F+F*F+F+F+F*" }{TEXT -1 37 " can be found using the substitution " } {XPPEDIT 18 0 "exp(x)=sinh*u" "6#/-%$expG6#%\"xG*&%%sinhG\"\"\"%\"uGF* " }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "u=arcsinh(exp(x))" "6#/% \"uG-%(arcsinhG6#-%$expG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)*sqrt(1+exp(2*x)),x)" "6#-%$I ntG6$*&-%$expG6#%\"xG\"\"\"-%%sqrtG6#,&F+F+-F(6#*&\"\"#F+F*F+F+F+F*" } {TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([exp(x)=sinh*u,``],[e xp(x)*dx=cosh*u*du,``])" "6#-%*PIECEWISEG6$7$/-%$expG6#%\"xG*&%%sinhG \"\"\"%\"uGF.%!G7$/*&-F)6#F+F.%#dxGF.*(%%coshGF.F/F.%#duGF.F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(sqrt(1+sinh^2*u)*cosh*u,u)" "6#/%!G-%$Int G6$*(-%%sqrtG6#,&\"\"\"F-*&%%sinhG\"\"#%\"uGF-F-F-%%coshGF-F1F-F1" } {XPPEDIT 18 0 "`` = Int(cosh^2*u,u)" "6#/%!G-%$IntG6$*&%%coshG\"\"#%\" uG\"\"\"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(``(1/2+cosh*2*u/2),u )" "6#/%!G-%$IntG6$-F$6#,&*&\"\"\"F,\"\"#!\"\"F,**%%coshGF,F-F,%\"uGF, F-F.F,F1" }{XPPEDIT 18 0 "``=u/2+sinh*2*u/4+c" "6#/%!G,(*&%\"uG\"\"\" \"\"#!\"\"F(**%%sinhGF(F)F(F'F(\"\"%F*F(%\"cGF(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "u +1/2" "6#,&%\"uG\"\"\"*&F%F%\"\"#!\"\"F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*u*cosh*u+c=1/2" "6#/,&**%%sinhG\"\"\"%\"uGF '%%coshGF'F(F'F'%\"cGF'*&F'F'\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u+1/2" "6#,&%\"uG\"\"\"*&F%F%\"\"#!\"\"F%" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sinh*u*sqrt(1+sinh^2*u)+c" "6#,&*(%%sinhG\"\"\"%\"uGF&- %%sqrtG6#,&F&F&*&F%\"\"#F'F&F&F&F&%\"cGF&" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsinh(exp(x))+1/2" "6#,&-%(arcsinhG6#-%$expG6#%\"xG\"\"\"*&F+F+\" \"#!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x)*sqrt(1+exp(2*x))+c " "6#,&*&-%$expG6#%\"xG\"\"\"-%%sqrtG6#,&F)F)-F&6#*&\"\"#F)F(F)F)F)F)% \"cGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\"\"\"F&\"\"# !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(exp(x)+sqrt(1+exp(2*x)))+1/2 " "6#,&-%#lnG6#,&-%$expG6#%\"xG\"\"\"-%%sqrtG6#,&F,F,-F)6#*&\"\"#F,F+F ,F,F,F,*&F,F,F4!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x)*sqrt(1+ exp(2*x))+c" "6#,&*&-%$expG6#%\"xG\"\"\"-%%sqrtG6#,&F)F)-F&6#*&\"\"#F) F(F)F)F)F)%\"cGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "The required surface area is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*Pi*Int(exp(x)*sqrt(1+e xp(2*x)),x=0..1)" "6#/%!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$*&-%$expG6#%\"xG F'-%%sqrtG6#,&F'F'-F.6#*&F&F'F0F'F'F'/F0;\"\"!F'F'" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*(ln(exp(x)+sq rt(1+exp(2*x)))+exp(x)*sqrt(1+exp(2*x)));" "6#/%!G*&%#PiG\"\"\",&-%#ln G6#,&-%$expG6#%\"xGF'-%%sqrtG6#,&F'F'-F.6#*&\"\"#F'F0F'F'F'F'*&-F.6#F0 F'-F26#,&F'F'-F.6#*&F8F'F0F'F'F'F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 " PIECEWISE([1,``],[``,``],[0,``])" "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F(F(7 $\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*(``(ln(exp(1)+sqrt(1+ex p(2)))+exp(1)*sqrt(1+exp(2)))-``(ln(1+sqrt(2))+sqrt(2)));" "6#/%!G*&%# PiG\"\"\",&-F$6#,&-%#lnG6#,&-%$expG6#F'F'-%%sqrtG6#,&F'F'-F16#\"\"#F'F 'F'*&-F16#F'F'-F46#,&F'F'-F16#F9F'F'F'F'-F$6#,&-F-6#,&F'F'-F46#F9F'F'- F46#F9F'!\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 390 1 "~" }{TEXT -1 15 " 22.9 4302234. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 172 "f := x -> exp(x):\n'f'(x)=f(x);\nDiff('f'(x),x)=di ff(f(x),x);\nsimplify(2*Pi*Int(f(x)*sqrt(1+rhs(%)^2),x=0..1));\n``=val ue(%);\n``=simplify(rhs(%));\n``=evalf(evalf[15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$phiG6#%\"xG-%$expGF&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%%DiffG6$-%$phiG6#%\"xGF*-%$expGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&-%$expG6#%\"xGF&, &F&F&-F-6#,$*&F%F&F/F&F&F&#F&F%/F/;\"\"!F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiGF(,**&F'!\"\"F'#F(F'F,*&#F(F'F( -%#lnG6#,&F(F(*$F'F-F(F(F,*&F-F(*&-%$expG6#F(F(,&F(F(-F86#F'F(F-F(F(*& F-F(-F16#,&F7F(*$F:F-F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G,$*&%#PiG\"\"\",**$\"\"##F(F+F(-%#lnG6#,&F(F(F*F(F(*&-%$expG6#F(F(,&F (F(-F36#F+F(F,!\"\"-F.6#,&F2F(*$F5F,F(F8F(F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+MAI%H#!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 6 " }}{PARA 0 "" 0 "" {TEXT 393 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 66 "Find the area of the su rface formed when the section of the curve " }{XPPEDIT 18 0 "y = 1/x; " "6#/%\"yG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 26 " from the the point whe re " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 20 " to the po int where " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" }{TEXT -1 22 " is \+ rotated about the " }{TEXT 395 1 "x" }{TEXT -1 8 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 394 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 572 "f := x -> 1/x:\np1 := plot([f(x),-f(x)],x=.5..4.3,y= -1.8..1.8,color=red,\n thickness=1,linestyle=[1,2]):\na := 1: b := 4 :\np2 := plot([f(x),-f(x)],x=a..b,\n color=[COLOR(RGB,.9,0,0),COLOR (RGB,.9,.7,.7)],thickness=3):\np3 := plot([[[a,f(a)],[b,f(b)]]$3],styl e=point,\n symbol=[circle,diamond,cross],color=black):\nh := evalf(Pi/20): r := .08:\np4 := plot([[seq([a+r*f(a)*cos(i*h),f(a)*sin (i*h)],i=0..40)],\n [seq([b+r*f(b)*cos(i*h),f(b)*sin(i*h)],i=0.. 40)]],\n linestyle=2,color=black):\nplots[display]([p1,p2,p3,p4] ,tickmarks=[5,3],view=[0..4.4,-1.8..1.8]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 255 255 {PLOTDATA 2 "6.-%'CURVESG6&7V7$$\"3++++++++]!#=$ \"\"#\"\"!7$$\"3:nm;Hf99aF*$\"3+Z(*3&Q8q%=!#<7$$\"3>LLLe=HGeF*$\"3w8k< s&odr\"F37$$\"3#***\\(o,Q')='F*$\"3KLdFBW'eh\"F37$$\"3mmmTvT)*[lF*$\"3 I,w7pV&p_\"F37$$\"3]+]7=6BapF*$\"3[veE\"\\tzV\"F37$$\"3ALL$31y%ftF*$\" 3'QW*y6?ze8F37$$\"3)HLLeUW`<)F*$\"3OO%f_0!>B7F37$$\"3AmmT?JL()*)F*$\"3 \"[Tlu7xE6\"F37$$\"3RLL3#e],u*F*$\"3*Gb_D0\"F3 $\"3;nT1w,-1&*F*7$$\"3?L$37$3eK6F3$\"38cL%[)=RH))F*7$$\"3!***\\P!=QH@ \"F3$\"3RH@1bKWW#)F*7$$\"3ymm;\"f&f&H\"F3$\"37P70!)yX=xF*7$$\"3KLLe$G+ %o8F3$\"3xj!45A.yI(F*7$$\"3-++]B6O]9F3$\"3g'\\N3cM[*oF*7$$\"3\"*****\\ -&eE`\"F3$\"31AIod1hClF*7$$\"3$*****\\ws'>h\"F3$\"3A;@5y)*f.iF*7$$\"3? 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" }}{PARA 0 "" 0 "" {TEXT -1 18 "If we assume that " }{TEXT 398 1 "x" }{TEXT -1 48 " is positive, the \+ last expression simplifies to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt(x^4+1)/(x^3);" "6#*&-%%sqrtG6#,&*$%\"xG\"\"%\"\"\" F+F+F+*$F)\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 "T he surface area is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*Int(f(x)*sqrt(1+`f '`(x)^2),x = 1 .. 4);" "6#*(\"\"#\"\"\"%#Pi GF%-%$IntG6$*&-%\"fG6#%\"xGF%-%%sqrtG6#,&F%F%*$-%$f~'G6#F.F$F%F%/F.;F% \"\"%F%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*Int(sqrt(1+x^4)/(x^3),x = 1 .. 4);" "6#/%!G*(\"\"#\"\" \"%#PiGF'-%$IntG6$*&-%%sqrtG6#,&F'F'*$%\"xG\"\"%F'F'*$F2\"\"$!\"\"/F2; F'F3F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "The indefinite integral " }{XPPEDIT 18 0 "Int(sqrt(1+x^4)/(x^3),x);" "6#-%$IntG6$*&- %%sqrtG6#,&\"\"\"F+*$%\"xG\"\"%F+F+*$F-\"\"$!\"\"F-" }{TEXT -1 37 " ca n be found using the substitution " }{XPPEDIT 18 0 "x^2 = sinh*u;" "6# /*$%\"xG\"\"#*&%%sinhG\"\"\"%\"uGF)" }{TEXT -1 11 ", that is, " } {XPPEDIT 18 0 "u = arcsinh(x^2);" "6#/%\"uG-%(arcsinhG6#*$%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "In t(sqrt(1+x^4)/(x^3),x);" "6#-%$IntG6$*&-%%sqrtG6#,&\"\"\"F+*$%\"xG\"\" %F+F+*$F-\"\"$!\"\"F-" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWI SE([x^2 = sinh*u, ``],[2*x*dx = cosh*u*du, ``],[x*dx = ``(cosh*u/2)*du , ``]);" "6#-%*PIECEWISEG6%7$/*$%\"xG\"\"#*&%%sinhG\"\"\"%\"uGF-%!G7$/ *(F*F-F)F-%#dxGF-*(%%coshGF-F.F-%#duGF-F/7$/*&F)F-F3F-*&-F/6#*(F5F-F.F -F*!\"\"F-F6F-F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(``(sqrt(1+x^4) /(x^4))*x,x)" "6#/%!G-%$IntG6$*&-F$6#*&-%%sqrtG6#,&\"\"\"F0*$%\"xG\"\" %F0F0*$F2F3!\"\"F0F2F0F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(``(sq rt(1+sinh^2*u)/(sinh^2*u))*``(cosh*u/2),u);" "6#/%!G-%$IntG6$*&-F$6#*& -%%sqrtG6#,&\"\"\"F0*&%%sinhG\"\"#%\"uGF0F0F0*&F2F3F4F0!\"\"F0-F$6#*(% %coshGF0F4F0F3F6F0F4" }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\" #!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cosh^2*u/(sinh^2*u),u) = 1 /2;" "6#/-%$IntG6$*(%%coshG\"\"#%\"uG\"\"\"*&%%sinhGF)F*F+!\"\"F**&F+F +F)F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(coth^2*u,u)" "6#-%$IntG6$*& %%cothG\"\"#%\"uG\"\"\"F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%! G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(1+csch^2 *u),u) = 1/2;" "6#/-%$IntG6$-%!G6#,&\"\"\"F+*&%%cschG\"\"#%\"uGF+F+F/* &F+F+F.!\"\"" }{XPPEDIT 18 0 "``(u-coth*u)+c = 1/2;" "6#/,&-%!G6#,&%\" uG\"\"\"*&%%cothGF*F)F*!\"\"F*%\"cGF**&F*F*\"\"#F-" }{XPPEDIT 18 0 "`` (u-sqrt(1+sinh^2*u)/(sinh*u))+c" "6#,&-%!G6#,&%\"uG\"\"\"*&-%%sqrtG6#, &F)F)*&%%sinhG\"\"#F(F)F)F)*&F0F)F(F)!\"\"F3F)%\"cGF)" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " \+ ``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(arcsinh(x^2)-s qrt(1+x^4)/x^2)+c=1/2" "6#/,&-%!G6#,&-%(arcsinhG6#*$%\"xG\"\"#\"\"\"*& -%%sqrtG6#,&F/F/*$F-\"\"%F/F/*$F-F.!\"\"F8F/%\"cGF/*&F/F/F.F8" } {XPPEDIT 18 0 "``(ln(x^2+sqrt(1+x^4))-sqrt(1+x^4)/x^2)+c" "6#,&-%!G6#, &-%#lnG6#,&*$%\"xG\"\"#\"\"\"-%%sqrtG6#,&F/F/*$F-\"\"%F/F/F/*&-F16#,&F /F/*$F-F5F/F/*$F-F.!\"\"F " 0 " " {MPLTEXT 1 0 142 "Int(sqrt(1+x^4)/x^3,x);\n``=value(%);\n``=map(norm al,[selectremove(has,rhs(%),arcsinh)]):\n``=op(2,rhs(%))+op(1,rhs(%)); \n``=convert(rhs(%),ln); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6 $*&%\"xG!\"$,&*$)F'\"\"%\"\"\"F-F-F-#F-\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"#!\"\"%\"xG!\"#,&*$)F)\"\"%\"\"\"F/F/F/#\" \"$F'F(*(F'F(F)F'F+#F/F'F/*&F3F/-%(arcsinhG6#*$)F)F'F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(\"\"#!\"\",&*$)%\"xG\"\"%\"\"\"F.F.F .#F.F'F,!\"#F(*&F/F.-%(arcsinhG6#*$)F,F'F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(\"\"#!\"\",&*$)%\"xG\"\"%\"\"\"F.F.F.#F.F'F,!\" #F(*&F/F.-%#lnG6#,&*$)F,F'F.F.*$F)F/F.F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "2*Pi*Int(sqrt(1+x^ 4)/x^3,x=1..4);\n``=value(%);\n``=evalf(evalf[15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xG!\"$,& *$)F,\"\"%F&F&F&F&#F&F%/F,;F&F1F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G,$*(\"\"#\"\"\"%#PiGF(,.*&#F(\"\"%F(-%#lnG6#,&\"#;!\"\"*$\"$d##F( F'F(F(F3*&#F(F-F(-F/6#,&F4F(F2F(F(F(*&\"#KF3F5F6F3*&F8F(-F/6#,&*$F'F6F (F(F3F(F(*&#F(F-F(-F/6#,&F(F(FBF(F(F3*&F'F3F'F6F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+&[PsT*!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 10 "Example 7 " }}{PARA 0 "" 0 "" {TEXT 383 8 "Questio n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 66 "Find the area of th e surface formed when the section of the curve " }{XPPEDIT 18 0 "y = 1 /x;" "6#/%\"yG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 26 " from the the point \+ where " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" }{TEXT -1 22 " \+ is rotated about the " }{TEXT 391 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT 384 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 646 "f := x -> 1 /x:\np1 := plot(f(x),x=.5..4.3,y=0..1.6,color=red,\n thickness=1,lin estyle=[1,2]):\np2 := plot(f(-x),x=-4.3..-0.5,y=0..1.6,color=red,\n \+ thickness=1,linestyle=[1,2]):\na := 1: b := 4:\np3 := plot(f(x),x=a..b ,color=COLOR(RGB,.9,0,0),thickness=3):\np4 := plot(f(-x),x=-b..-a,colo r=COLOR(RGB,.9,.7,.7),thickness=3):\np5 := plot([[[a,f(a)],[b,f(b)]]$3 ],style=point,\n symbol=[circle,diamond,cross],color=black): \nh := evalf(Pi/20): \np6 := plot([[seq([a*cos(i*h),f(a)+.03*a*sin(i*h )],i=0..40)],\n [seq([b*cos(i*h),f(b)+.03*b*sin(i*h)],i=0..40)]] ,\n linestyle=2,color=black):\nplots[display]([p1,p2,p3,p4,p5,p6 ],tickmarks=[5,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 680 279 279 {PLOTDATA 2 "6.-%'CURVESG6&7V7$$\"3++++++++]!#=$\"\"#\"\"!7$$\"3:nm;Hf 99aF*$\"3+Z(*3&Q8q%=!#<7$$\"3>LLLe=HGeF*$\"3w8kB7F37$$\"3AmmT?JL()*)F*$\"3\"[Tlu7xE6\"F37$$\"3RL L3#e],u*F*$\"3*Gb_D0\"F3$\"3;nT1w,-1&*F*7$$\"3 ?L$37$3eK6F3$\"38cL%[)=RH))F*7$$\"3!***\\P!=QH@\"F3$\"3RH@1bKWW#)F*7$$ \"3ymm;\"f&f&H\"F3$\"37P70!)yX=xF*7$$\"3KLLe$G+%o8F3$\"3xj!45A.yI(F*7$ $\"3-++]B6O]9F3$\"3g'\\N3cM[*oF*7$$\"3\"*****\\-&eE`\"F3$\"31AIod1hClF *7$$\"3$*****\\ws'>h\"F3$\"3A;@5y)*f.iF*7$$\"3?L$37C()Ro\"F3$\"3$*zufr sGQfF*7$$\"3smm;XfipF3$\"3'))>&y#>\"f!>&F*7$$\"3fmmml>E,?F3$\"3;5ZFvq% o*\\F*7$$\"3o**\\()o(=K3#F3$\"3%f,G5(RE+[F*7$$\"33+]iE5Eh@F3$\"3b>'R*) QGpi%F*7$$\"3tmmTN**oUAF3$\"3?lKGW6$*eWF*7$$\"3am;/$4nuJ#F3$\"3'4qzr(e 0:VF*7$$\"3-LLefV7)R#F3$\"3O`RpLb#*pTF*7$$\"33L$37f/>[#F3$\"39&=Hguj\" HSF*7$$\"3++]7Hb$[b#F3$\"3krP\"pGYT\"RF*7$$\"3=LL3SHgLEF3$\"3%38\\1gzq z$F*7$$\"3\"******HQx\\r#F3$\"3E<F$o$F*7$$\"3c***\\(*R'e%z#F3$\"3S fL<]qMyNF*7$$\"3++](o@7;(GF3$\"3wL]9YVO#[$F*7$$\"3#****\\UDOr&HF3$\"3a &HOG1];Q$F*7$$\"38mmm0M)R.$F3$\"3Yl$\\I$p*fH$F*7$$\"3x****\\UT.;JF3$\" 3i&H`\\X2#4KF*7$$\"39L$3_I%Q!>$F3$\"3i')o-H'=W8$F*7$$\"33++]r'o;F$F3$ \"3o@X*)HUacIF*7$$\"3Qm;aD4:[LF3$\"39'fOh=Bn)HF*7$$\"33+]P=p4GMF3$\"3: P)GdQqq\"HF*7$$\"3Umm;R(ei]$F3$\"3%*)3)G-G/_GF*7$$\"3c**\\(=#p3)e$F3$ \"3%eNst[+qy#F*7$$\"3OLLL(=(*om$F3$\"3rE))Rw75FFF*7$$\"3MLLem?\\ZPF3$ \"3yiD]z7XoEF*7$$\"3cm;az&>u#QF3$\"3X^>9`js7EF*7$$\"37+++jY'3!RF3$\"3# eXxo?MNc#F*7$$\"3emm\"p,T])RF3$\"3.'Q@*fWQ4DF*7$$\"3SLLL^$H.1%F3$\"3== g$zRaGY#F*7$$\"3\"***\\()=CgSTF3$\"3i(fV0^2^T#F*7$$\"3<+]7()RV_S1%F3$\"3]9q\")>$)fgCF*7$$!3_mmT dbY#)RF3$\"3FVmBys+6DF*7$$!34L$ezom7!RF3$\"3WBX0N+FjDF*7$$!3[m;zT\\)f# QF3$\"33ahs_gq8EF*7$$!3#)**\\7a].[PF3$\"3_&GdSpk!oEF*7$$!3im;zo\">um$F 3$\"38OLzHJrEFF*7$$!39+]i>=1(e$F3$\"3m\"*3%*\\pz(y#F*7$$!3/LL$)3WS/NF3 $\"3)R^z,%>b`GF*7$$!3smmT;(*fJMF3$\"3e#f'R#z#49HF*7$$!3!)****\\w)Q'\\L F3$\"3Ux+\"oW'R&)HF*7$$!3#*****\\(\\TtE$F3$\"3/]%)**HAfgIF*7$$!35++]BF .)=$F3$\"3>Si[j1tOJF*7$$!3Rm;zeF,;JF3$\"3s[B@y%H#4KF*7$$!36LL$[0u..$F3 $\"3#RH*4cE#**H$F*7$$!3MLLLRo#y&HF3$\"3;?**pC0'3Q$F*7$$!3!***\\7\"=PM( GF3$\"3!RWlGk_,[$F*7$$!3CLLLM!Q()z#F3$\"3kvKJZ*QId$F*7$$!39+]7J7y;FF3$ \"3-Ht3rj#3o$F*7$$!3u**\\Pt*Q(QEF3$\"3'[P!yw*)o*y$F*7$$!35LLek+JdDF3$ \"3f#e#4L)e.\"RF*7$$!3GL$ep!H`#[#F3$\"3b'pF3$\"3EIS-%)*yc=&F*7$$!3*)***\\dujG%=F3$\"3@\"eqn9PjU&F*7$$!3qLLL%f; gw\"F3$\"3y!\\/'[4YicF*7$$!3/++]de'Ro\"F3$\"3=NDZyEOQfF*7$$!3mm;z%p:'4 ;F3$\"3UP\"F3$\"3o!GaVkW\"*G(F*7$$!3SLL$3 ETPH\"F3$\"30@\"F3$\"3+:2IboT^#)F*7$$!3Ymmm7G5 L6F3$\"3\"[#p&z[C`#))F*7$$!3[mmTLz]_5F3$\"3BOt+Kg6,&*F*7$$!3iKLe/U!es* F*$\"33!QF!4E>G5F37$$!3-(*****pLN\"**)F*$\"3R2Z&Giz@6\"F37$$!3aKL$3$)* e\\\")F*$\"3uZ'y'>c0F7F37$$!3uomm'[1nR(F*$\"3Ot!\\U1`>N\"F37$$!3!RLe*[ 6M&*pF*$\"3-_arZG_H9F37$$!35***\\7\"e(Rf'F*$\"3ByR0Ud`;:F37$$!3-+++qz \")4iF*$\"3YfS*\\\"F3$\" 3%e9Q623$pmF*7$$\"35+](=$f%Gc\"F3$\"3?/`nwLe)R'F*7$$\"3$*****\\#y,\"G; F3$\"3P*[cs8A@9'F*7$$\"37++Dr\"zbo\"F3$\"3+pmv4*yE$fF*7$$\"3%*****\\(4 &G]F3$\"3'3U&=)\\)oo^F*7$$\"3,++]s iL-?F3$\"3=N4]Lh;%*\\F*7$$\"3-+++!R5'f?F3$\"3#ed&pSG>ahK%F*7$$ \"33++vVy!eP#F3$\"3\\6&4Wh%44UF*7$$\"3K+](=WU[V#F3$\"32e=HH>/2TF*7$$\" 3)****\\7B>&)\\#F3$\"3x$QWTjqB+%F*7$$\"3)***\\P>:mkDF3$\"3-w6K9+:**QF* 7$$\"3'***\\iv&QAi#F3$\"3uTZ/Rb`8QF*7$$\"31++vtLU%o#F3$\"3$[)yk-[>DPF* 7$$\"37+++bjm[FF3$\"3#f@G&*)z7QOF*7$$\"3\"****\\(yb^6GF3$\"3ES\"=U\\+o b$F*7$$\"3)***\\PMaKsGF3$\"3iCy#>l*\\\"[$F*7$$\"3=++D6W%)RHF3$\"30ITJe 0a,MF*7$$\"3z*****\\@80+$F3$\"3Yw%z>>jFL$F*7$$\"31++]7,HlIF3$\"3QF;!) \\RLiKF*7$$\"3()**\\P4w)R7$F3$\"35n%fR-P5?$F*7$$\"3;++]x%f\")=$F3$\"3Z /%zT&fgOJF*7$$\"3!)**\\P/-a[KF3$\"3caj+`dIyIF*7$$\"3/+](=Yb;J$F3$\"3() 3x*fzP'>IF*7$$\"3')****\\i@OtLF3$\"3)ox9Ja,W'HF*7$$\"3')**\\PfL'zV$F3$ \"3ZA+rH))p3HF*7$$\"3>+++!*>=+NF3$\"3S9?g,V*p&GF*7$$\"3-++DE&4Qc$F3$\" 3bWjw,h)f!GF*7$$\"3=+]P%>5pi$F3$\"3M&z2'*znrv#F*7$$\"39+++bJ*[o$F3$\"3 !ob;7*Gy8FF*7$$\"33++Dr\"[8v$F3$\"3'*e:l6$3dm#F*7$$\"3++++Ijy5QF3$\"3^ FAqK.8CEF*7$$\"31+]P/)fT(QF3$\"3/$)\\>I[?\"e#F*7$$\"31+]i0j\"[$RF3$\"3 0Vuw*p99a#F*7$$\"\"%F-$\"3++++++++DF*-%&COLORG6&F_\\l$\"\"*!\"\"Fc\\lF c\\l-Fe\\l6#\"\"$-F$6%7S7$$!\"%F-F\\\\n7$$!3&*****\\P&3Y$RF3$\"3nZ[*)* *)[:a#F*7$$!3!***\\iv! 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z*pS%3%o%F/$!3EG-YW/7N))F/Ffffl7%$\"3-y#=y%G'QZ'F/$!3SijF^0i@wF/Ffffl7 %$\"3wRk0d1$4'zF/$!3)zkP>:U<0'F/Ffffl7%$\"3'*o&4n>ad2*F/$!3K\\Eg:5*))> %F/Ffffl7%Fiffl$!31W-6-Wq\\@F/Ffffl7%Ffffl$!3%\\q%H)f$H\\CFhtFfffl-%&S TYLEG6#%,PATCHNOGRIDG-%'CURVESG6%7$7%F,F,F,7%F,F,$\"#:!\"\"-%'COLOURG6 &%$RGBG$\")#)eqk!\")$\"))eqk\"Fh`glFi`gl-Ff_gl6#%%LINEG-%+LIGHTMODELG6 #%(LIGHT_2G-%*AXESTICKSG6%\"\"$FeaglFeagl-%*AXESSTYLEG6#%&FRAMEG-%+AXE SLABELSG6%%\"xG%!G%\"yG-%+PROJECTIONG6%$!$5\"F+$\"#]F+Fgffl" 1 2 0 1 10 0 2 1 4 3 2 1.000000 50.000000 -110.000000 1 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "f( x) = 1/x;" "6#/-%\"fG6#%\"xG*&\"\"\"F)F'!\"\"" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "`f '`(x) = -1/(x^2);" "6#/-%$f~'G6#%\"xG,$*&\"\"\"F **$F'\"\"#!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*sqrt(1+`f '`(x) ^2) = x*sqrt(1+1/(x^4));" "6#/*&%\"xG\"\"\"-%%sqrtG6#,&F&F&*$-%$f~'G6# F%\"\"#F&F&*&F%F&-F(6#,&F&F&*&F&F&*$F%\"\"%!\"\"F&F&" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 18 "If we assume that " }{TEXT 397 1 "x" }{TEXT -1 48 " is positive, the last expression simplifies to " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x^4+1)/x;" "6#*& -%%sqrtG6#,&*$%\"xG\"\"%\"\"\"F+F+F+F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 "The surface area is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*Int(f(x)*sqrt(1+`f '`(x)^2),x = 1 \+ .. 4);" "6#*(\"\"#\"\"\"%#PiGF%-%$IntG6$*&-%\"fG6#%\"xGF%-%%sqrtG6#,&F %F%*$-%$f~'G6#F.F$F%F%/F.;F%\"\"%F%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*Int(sqrt(1+x^4)/(x^3),x = \+ 1 .. 4);" "6#/%!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$*&-%%sqrtG6#,&F'F'*$%\"x G\"\"%F'F'*$F2\"\"$!\"\"/F2;F'F3F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 24 "The indefinite integral " }{XPPEDIT 18 0 "Int(sqrt(1+x^ 4)/x,x);" "6#-%$IntG6$*&-%%sqrtG6#,&\"\"\"F+*$%\"xG\"\"%F+F+F-!\"\"F- " }{TEXT -1 37 " can be found using the substitution " }{XPPEDIT 18 0 "1+x^4 = u^2;" "6#/,&\"\"\"F%*$%\"xG\"\"%F%*$%\"uG\"\"#" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "u = sqrt(1+x^4);" "6#/%\"uG-%%sqrtG6#,& \"\"\"F)*$%\"xG\"\"%F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "u>=1" "6#1\"\"\"%\"uG" }{TEXT -1 2 ". \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(1+x^4)/( x),x);" "6#-%$IntG6$*&-%%sqrtG6#,&\"\"\"F+*$%\"xG\"\"%F+F+F-!\"\"F-" } {TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([1+x^4 = u^2, x^4 = u ^2-1],[4*x^3*dx = 2*u*du, ``],[x^3*dx = ``(u/2)*du, ``]);" "6#-%*PIECE WISEG6%7$/,&\"\"\"F)*$%\"xG\"\"%F)*$%\"uG\"\"#/*$F+F,,&*$F.F/F)F)!\"\" 7$/*(F,F)*$F+\"\"$F)%#dxGF)*(F/F)F.F)%#duGF)%!G7$/*&F+F9F:F)*&-F=6#*&F .F)F/F4F)F " 0 "" {MPLTEXT 1 0 130 "Int(sqrt(1+x^4)/x,x);\n``=value(%) ; \n``=convert(rhs(%),ln);\n``=1/4*combine(4*rhs(%),ln,integer,symboli c);\n``=map(simplify,rhs(%)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$I ntG6$*&,&\"\"\"F(*$)%\"xG\"\"%F(F(#F(\"\"#F+!\"\"F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,&*&\"\"#!\"\",&\"\"\"F**$)%\"xG\"\"%F*F*#F*F'F** &#F*F'F*-%(arctanhG6#*&F*F**$F)#F*F'F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&\"\"#!\"\",&\"\"\"F**$)%\"xG\"\"%F*F*#F*F'F**&# F*F.F*-%#lnG6#,&*&F*F**$F)#F*F'F(F*F*F*F*F(*&#F*F.F*-F36#,&F*F*F6F(F*F *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"#!\"\",&\"\"\"F**$)% \"xG\"\"%F*F*#F*F'F**&#F*F.F*-%#lnG6#*&,&*&F*F**$F)#F*F'F(F*F*F*F(,&F* F*F7F(F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"#!\"\",&\" \"\"F**$)%\"xG\"\"%F*F*#F*F'F**&#F*F.F*-%#lnG6#*&,&*$F)F/F*F*F(F*,&F*F *F7F*F(F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "2*Pi*Int(sqrt(1+x^4)/x,x=1..4);\n``=value(%);\n``= evalf(evalf[15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"# \"\"\"%#PiGF&-%$IntG6$*&,&F&F&*$)%\"xG\"\"%F&F&#F&F%F/!\"\"/F/;F&F0F&F &" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiGF(,.*&F'! \"\"\"$d##F(F'F(*&#F(\"\"%F(-%#lnG6#,&*$F-F.F(F(F,F(F(*&#F(F1F(-F36#,& F6F(F(F(F(F,*&F'F,F'F.F,*&#F(F1F(-F36#,&*$F'F.F(F(F,F(F,*&F0F(-F36#,&F (F(FBF(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+YtL\\[!\") " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 8 \+ " }}{PARA 0 "" 0 "" {TEXT 400 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 66 "Find the area of the surface formed when the sect ion of the curve " }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$sinG\"\"\" %\"xGF'" }{TEXT -1 26 " from the the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 22 " is rotated about the " } {TEXT 402 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 401 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 553 "f := x -> sin(x):\np1 := plot([f(x),-f(x)],x=-.2..3.5,y=-1.2..1.2,color=re d,\n thickness=1,linestyle=[1,2]):\na := 0: b := Pi:\np2 := plot([f( x),-f(x)],x=a..b,\n color=[COLOR(RGB,.9,0,0),COLOR(RGB,.9,.7,.7)],t hickness=3):\np3 := plot([[[a,f(a)],[b,f(b)]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\nh := evalf(Pi/20): r \+ := .08:\np4 := plot([[seq([a+r*f(a)*cos(i*h),f(a)*sin(i*h)],i=0..40)], \n [seq([b+r*f(b)*cos(i*h),f(b)*sin(i*h)],i=0..40)]],\n li nestyle=2,color=black):\nplots[display]([p1,p2,p3,p4],tickmarks=[3,5]) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 450 270 270 {PLOTDATA 2 "6.-%'CURVESG6 &7S7$$!35+++++++?!#=$!3;71&zI$p')>F*7$$!3]LL$eH0N>\"F*$!3Q0%yC#Qn!>\"F *7$$!3KlmTgcy<\\!#>$!3O+c:aO!e\"\\F57$$\"3qmm;H`'Q(HF5$\"31zc*p+FM(HF5 7$$\"3;nmTIFy\"4\"F*$\"3av\"\\M.:'*3\"F*7$$\"3+L$e9I.C)=F*$\"3/FAC-gIr =F*7$$\"3Cm;Hx(4ah#F*$\"3;%4N]J%p&e#F*7$$\"3o**\\iDbRuLF*$\"3-n&)ey7s5 LF*7$$\"3-n;Hs'R$fTF*$\"38MOlIWWSSF*7$$\"3%***\\7$\\m<%\\F*$\"3))*4r\\ npIu%F*7$$\"3/LL$3l)eYdF*$\"3'>!>\\7[[NaF*7$$\"3)zm;aWwaX'F*$\"3PTqj4( fj,'F*7$$\"31,+]-i^`sF*$\"3Z8$y@*e)Rj'F*7$$\"3U,+](yK[0)F*$\"3.cKRZ_l6 sF*7$$\"3u,+](z\\q#))F*$\"3\"fqg-Q&fCxF*7$$\"3Tm;HAoHG&*F*$\"3cF'fMC#e ]\")F*7$$\"3PLL3OZ@O5!#<$\"3Y(=I>w=[g)F*7$$\"3QLLL9G&o5\"Ffp$\"3*[HPz> [H%*)F*7$$\"33+D1#p@!*=\"Ffp$\"3gEXR_#\\+G*F*7$$\"3CLL$=2b+D1<^bT8Ffp$\"3*R(\\9=5RQ(*F*7$$\"3E+v=JOa<9Ffp$\"3a %)\\F9xz#))*F*7$$\"3aL$eRnHo\\\"Ffp$\"3E9m'49dE(**F*7$$\"3kLek6!R'p:Ff p$\"2g(*3JI$******Ffp7$$\"3sm;a=P<[;Ffp$\"3?4JiH'y+(**F*7$$\"3qm\"Hs? \\(HFfp$\"3?y!>P)*HYE*F*7 $$\"3/+]7ZD?M?Ffp$\"3EXuNs6NX*)F*7$$\"3y*\\i!p8?4@Ffp$\"3'oHMB:w_e)F*7 $$\"3))**\\(QxuC>#Ffp$\"3Z!Q\"HR2-H\")F*7$$\"3;LL$=j*HnAFfp$\"3'omZiR, 4n(F*7$$\"3'****\\(Q6>ZBFfp$\"3n9BsqOWMrF*7$$\"3Km\"H#=Ze>CFfp$\"3j@Q5 !>H*3mF*7$$\"3=++DA+t)\\#Ffp$\"3mr(ex&4*[*fF*7$$\"3%H$eR&e*>tDFfp$\"3+ &[[8v%y#Q&F*7$$\"31+DJO6r%F*7$$\"3JLL3nm9FFFfp$\"3]g 6Y<)Go-%F*7$$\"3#**\\ilZ@o!GFfp$\"3q[6%\\eIbG$F*7$$\"3qmmm(ydN)GFfp$\" 3+KPY3(4=b#F*7$$\"3am;/\\<.iHFfp$\"3_p'**)[b(fy\"F*7$$\"3qLeR1f&)RIFfp $\"3OY&o!fLh:5F*7$$\"31++]C#o86$Ffp$\"3Rfw%\\v#)>-$F57$$\"3YL$37THL>$F fp$!31C%oX,o8<&F57$$\"3)ommOZOmE$Ffp$!3eacV9@=Z7F*7$$\"33+D1#4(zWLFfp$ !3)ze$e%o)3=?F*7$$\"3/+v$px1'>MFfp$!3l\"=&HtcYWFF*7$$\"3++++++++NFfp$! 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3'*))o/b+VIn(**F*7$Fi\\m$!274'Gxz)*****Ffp7$F^]m$!39vB0ZK3x**F*7$Fc] m$!3[oG=d;==**F*7$Fh]m$!3Vz\\$>Q(39)*F*7$F]^m$!3?VTu'[jGm*F*7$Fb^m$!3S m0JN*4EZ*F*7$Fg^m$!3ADf%RFx%\\#*F*7$F\\_m$!3?af@Y>%y&*)F*7$Fa_m$!3?%>' G5ccd')F*7$Ff_m$!3-mo\">-G%)H)F*7$F[`m$!3cJr!4)G)*RzF*7$F``m$!3l!*y#R$ G%Q^(F*7$Fe`m$!3'4._Q1p=3(F*7$Fj`m$!3BSu)y%[5+mF*7$F_am$!3_nY*\\Y775'F *7$Fdam$!3OF*7$Fgcm$!3M.OA\"o%)RJ\"F*7$F\\dm$!3ls_wI6s?oF57$Fadm$!3pawpOM zRJFedm-Fgdm6&F`[lFidm$\"\"(F[emFd^nF\\em-F$6&7$Fdel7$$\"37$z*e`EfTJFf pFd[l-%'SYMBOLG6#%'CIRCLEG-F^[l6&F`[lFe[lFe[lFe[l-%&STYLEG6#%&POINTG-F $6&Fh^n-F]_n6#%(DIAMONDGF`_nFb_n-F$6&Fh^n-F]_n6#%&CROSSGF`_nFb_n-F$6%7 KFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelF delFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFdelFde lFdelFdelFdelFdelFdelFdelF`_nF^el-F$6%7KFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi ^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^n Fi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nFi^nF`_nF^ el-%*AXESTICKSG6$F^em\"\"&-%+AXESLABELSG6%Q\"x6\"Q\"yF^an-%%FONTG6#%(D EFAULTG-%%VIEWG6$;$!\"#F[em$\"#NF[em;$!#7F[em$\"#7F[em" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "f(x) = s in*x;" "6#/-%\"fG6#%\"xG*&%$sinG\"\"\"F'F*" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "`f '`(x) = cos*x;" "6#/-%$f~'G6#%\"xG*&%$cosG\"\"\"F'F *" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)*sqrt(1+`f '`(x)^2) = sin*x; " "6#/*&-%\"fG6#%\"xG\"\"\"-%%sqrtG6#,&F)F)*$-%$f~'G6#F(\"\"#F)F)*&%$s inGF)F(F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1+cos^2*x);" "6#-%%sqr tG6#,&\"\"\"F'*&%$cosG\"\"#%\"xGF'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "The surface area is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*Int(f(x)*sqrt(1+`f '`(x)^2),x = 0 .. Pi);" "6 #*(\"\"#\"\"\"%#PiGF%-%$IntG6$*&-%\"fG6#%\"xGF%-%%sqrtG6#,&F%F%*$-%$f~ 'G6#F.F$F%F%/F.;\"\"!F&F%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*Int(sin*x*sqrt(1+cos^2*x),x = 0 .. \+ Pi);" "6#/%!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$*(%$sinGF'%\"xGF'-%%sqrtG6#, &F'F'*&%$cosGF&F.F'F'F'/F.;\"\"!F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "The indefinite integral " }{XPPEDIT 18 0 "Int(sin*x*sq rt(1+cos^2*x),x);" "6#-%$IntG6$*(%$sinG\"\"\"%\"xGF(-%%sqrtG6#,&F(F(*& %$cosG\"\"#F)F(F(F(F)" }{TEXT -1 37 " can be found using the substitut ion " }{XPPEDIT 18 0 "cos*x = sinh*u;" "6#/*&%$cosG\"\"\"%\"xGF&*&%%si nhGF&%\"uGF&" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "u=arcsinh(cos *x)" "6#/%\"uG-%(arcsinhG6#*&%$cosG\"\"\"%\"xGF*" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*x*sqrt(1+cos^ 2*x),x);" "6#-%$IntG6$*(%$sinG\"\"\"%\"xGF(-%%sqrtG6#,&F(F(*&%$cosG\" \"#F)F(F(F(F)" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([cos* x = sinh*u, ``],[-sin*x*dx = cosh*u*du, ``],[sin*x*dx = -cosh*u*du, `` ]);" "6#-%*PIECEWISEG6%7$/*&%$cosG\"\"\"%\"xGF**&%%sinhGF*%\"uGF*%!G7$ /,$*(%$sinGF*F+F*%#dxGF*!\"\"*(%%coshGF*F.F*%#duGF*F/7$/*(F4F*F+F*F5F* ,$*(F8F*F.F*F9F*F6F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -Int(cosh*u*sq rt(1+sinh*u^2),u);" "6#/%!G,$-%$IntG6$*(%%coshG\"\"\"%\"uGF+-%%sqrtG6# ,&F+F+*&%%sinhGF+*$F,\"\"#F+F+F+F,!\"\"" }{XPPEDIT 18 0 "`` = -Int(cos h^2*u,u);" "6#/%!G,$-%$IntG6$*&%%coshG\"\"#%\"uG\"\"\"F,!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -Int(``(1/2+cosh*2*u/2),u);" "6#/%!G,$-%$In tG6$-F$6#,&*&\"\"\"F-\"\"#!\"\"F-**%%coshGF-F.F-%\"uGF-F.F/F-F2F/" } {XPPEDIT 18 0 "`` = -u/2-sinh*2*u/4+c;" "6#/%!G,(*&%\"uG\"\"\"\"\"#!\" \"F***%%sinhGF(F)F(F'F(\"\"%F*F*%\"cGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = -1/2;" "6#/%!G,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "u-1/2;" "6#,&%\"uG\"\"\"*&F%F%\"\"#!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*u*cosh*u+c = -1/2;" "6#/,&**%%sinhG\"\"\"% \"uGF'%%coshGF'F(F'F'%\"cGF',$*&F'F'\"\"#!\"\"F." }{TEXT -1 1 " " } {XPPEDIT 18 0 "u-1/2;" "6#,&%\"uG\"\"\"*&F%F%\"\"#!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*u*sqrt(1+sinh^2*u)+c" "6#,&*(%%sinhG\"\"\"% \"uGF&-%%sqrtG6#,&F&F&*&F%\"\"#F'F&F&F&F&%\"cGF&" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = -1/2;" "6#/%!G,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsinh(cos*x)-1/2;" "6#,&-%(arcsinhG6#*&%$cosG \"\"\"%\"xGF)F)*&F)F)\"\"#!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos *x*sqrt(1+cos^2*x)+c;" "6#,&*(%$cosG\"\"\"%\"xGF&-%%sqrtG6#,&F&F&*&F% \"\"#F'F&F&F&F&%\"cGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/2;" "6# /%!G,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(cos*x +sqrt(1+cos^2*x))-1/2;" "6#,&-%#lnG6#,&*&%$cosG\"\"\"%\"xGF*F*-%%sqrtG 6#,&F*F**&F)\"\"#F+F*F*F*F**&F*F*F1!\"\"F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x*sqrt(1+cos^2*x)+c;" "6#,&*(%$cosG\"\"\"%\"xGF&-%%sqrtG6#,& F&F&*&F%\"\"#F'F&F&F&F&%\"cGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The required surface area is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*Int (sin*x*sqrt(1+cos^2*x),x = 0 .. Pi);" "6#/%!G*(\"\"#\"\"\"%#PiGF'-%$In tG6$*(%$sinGF'%\"xGF'-%%sqrtG6#,&F'F'*&%$cosGF&F.F'F'F'/F.;\"\"!F(F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= -Pi*(ln(cos*x+sqrt(1+cos^2*x))+cos*x*sqrt(1+cos^2*x))" "6#/%!G,$*&%#Pi G\"\"\",&-%#lnG6#,&*&%$cosGF(%\"xGF(F(-%%sqrtG6#,&F(F(*&F/\"\"#F0F(F(F (F(*(F/F(F0F(-F26#,&F(F(*&F/F6F0F(F(F(F(F(!\"\"" }{TEXT -1 2 " " } {XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG 6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-Pi*(ln( sqrt(2)-1)-sqrt(2))+Pi*(ln(1+sqrt(2))+sqrt(2))" "6#/%!G,&*&%#PiG\"\"\" ,&-%#lnG6#,&-%%sqrtG6#\"\"#F(F(!\"\"F(-F/6#F1F2F(F2*&F'F(,&-F+6#,&F(F( -F/6#F1F(F(-F/6#F1F(F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Pi*(2*sq rt(2)-ln(sqrt(2)-1)+ln(1+sqrt(2)));" "6#/%!G*&%#PiG\"\"\",(*&\"\"#F'-% %sqrtG6#F*F'F'-%#lnG6#,&-F,6#F*F'F'!\"\"F4-F/6#,&F'F'-F,6#F*F'F'F'" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Pi*(2*sqrt(2)+ln(1/(sqrt(2)-1))+l n(1+sqrt(2)));" "6#/%!G*&%#PiG\"\"\",(*&\"\"#F'-%%sqrtG6#F*F'F'-%#lnG6 #*&F'F',&-F,6#F*F'F'!\"\"F5F'-F/6#,&F'F'-F,6#F*F'F'F'" }{TEXT -1 2 " \+ \n" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 2*Pi*(sqr t(2)+ln(1+sqrt(2)));" "6#/%!G*(\"\"#\"\"\"%#PiGF',&-%%sqrtG6#F&F'-%#ln G6#,&F'F'-F+6#F&F'F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 403 1 "~" }{TEXT -1 15 " \+ 14.42359945. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "Int(sin(x)*sqrt(1+cos(x)^2),x);\n``=student[cha ngevar](cos(x)=sinh(u),%,u);\n``=simplify(rhs(%),symbolic);\n``=value( rhs(%))+c;\n``=eval(subs(u=arcsinh(cos(x)),rhs(%))); \n``=convert(rhs( %),ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$sinG6#%\"xG\" \"\",&F+F+*$)-%$cosGF)\"\"#F+F+#F+F1F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,$*&,&\"\"\"F+*$)-%%sinhG6#%\"uG\"\"#F+F+#F+F2-%%co shGF0F+!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$-%$IntG6$*$)-% %coshG6#%\"uG\"\"#\"\"\"F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G,(*&#\"\"\"\"\"#F(*&-%%coshG6#%\"uGF(-%%sinhGF-F(F(!\"\"*&F)F1F.F(F1% \"cGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\"\"\"\"\"#F(*&,&F( F(*$)-%$cosG6#%\"xGF)F(F(#F(F)F.F(F(!\"\"*&#F(F)F(-%(arcsinhG6#F.F(F3% \"cGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\"\"\"\"\"#F(*&,&F( F(*$)-%$cosG6#%\"xGF)F(F(#F(F)F.F(F(!\"\"*&#F(F)F(-%#lnG6#,&F.F(*$F+F2 F(F(F3%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 84 "2*Pi*Int(sin(x)*sqrt(1+cos(x)^2),x=0..Pi);\n`` =value(%);\n``=evalf(evalf[15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&-%$sinG6#%\"xGF&,&F&F&*$)-%$cos GF.F%F&F&#F&F%/F/;\"\"!F'F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, $*(\"\"#\"\"\"%#PiGF(,&*$F'#F(F'F(-%#lnG6#,&F(F(F+F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+X*fBW\"!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 " Q1 " }}{PARA 0 "" 0 "" {TEXT -1 66 "Find the area of the surface forme d when the section of the curve " }{XPPEDIT 18 0 "y = sqrt(1-x^2);" "6 #/%\"yG-%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#!\"\"" }{TEXT -1 15 " from the \+ point" }{XPPEDIT 18 0 "``(-1/sqrt(2),1/sqrt(2))" "6#-%!G6$,$*&\"\"\"F( -%%sqrtG6#\"\"#!\"\"F-*&F(F(-F*6#F,F-" }{TEXT -1 13 " to the point" } {XPPEDIT 18 0 "``(1/sqrt(2),1/sqrt(2))" "6#-%!G6$*&\"\"\"F'-%%sqrtG6# \"\"#!\"\"*&F'F'-F)6#F+F," }{TEXT -1 22 " is rotated about the " } {TEXT 385 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 359 "f := x -> sqrt(1- x^2):\n'f(x)'=f(x);\np1 := plots[tubeplot]([x,0,0],x=-1/sqrt(2)..1/sqr t(2),radius=f(x),tubepoints=30,\n style=patchnogrid):\np2 := \+ plots[pointplot3d]([[-1.8,0,0],[1.8,0,0]],style=line,color=brown):\npl ots[display]([p1,p2],labels=[`x`,``,`y`],lightmodel=light4,\n scali ng=constrained,axes=framed,tickmarks=[3,3,3],orientation=[-120,60]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "f := x -> sqrt(1-x^2):\n'f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x ),x);\n2*Pi*Int(f(x)*sqrt(1+D(f)(x)^2),x=-1/sqrt(2)..1/sqrt(2));\n``=s implify(%,symbolic);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$,&\"\"\"F**$)F'\"\"#F*!\"\"#F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,$*&,&\"\"\"F.*$)F*\"\"# F.!\"\"#F2F1F*F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"% #PiGF&-%$IntG6$*&,&F&F&*$)%\"xGF%F&!\"\"#F&F%,&F&F&*&F,F0F/F%F&F1/F/;, $*&F%F0F%F1F0,$*&F%F0F%F1F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G,$*(\"\"#\"\"\"%#PiGF(-%$IntG6$F(/%\"xG;,$*&F'!\"\"F'#F(F'F2,$*&F'F2F 'F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiGF (F'#F(F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*Int(1,x = -1/sqrt(2) .. 1/ sqrt(2)) = 2*sqrt(2)*Pi;" "6#/*(\"\"#\"\"\"%#PiGF&-%$IntG6$F&/%\"xG;,$ *&F&F&-%%sqrtG6#F%!\"\"F3*&F&F&-F16#F%F3F&*(F%F&-F16#F%F&F'F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_________________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 66 "Find the are a of the surface formed when the section of the curve " }{XPPEDIT 18 0 "y = cosh*x;" "6#/%\"yG*&%%coshG\"\"\"%\"xGF'" }{TEXT -1 22 " from t he point where " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" } {TEXT -1 22 " is rotated about the " }{TEXT 386 1 "x" }{TEXT -1 7 " ax is. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 136 "f := x -> cosh(x):\n'f'(x)=f(x);\nDiff('f'(x),x)=d iff(f(x),x);\n2*Pi*Int(f(x)*sqrt(1+D(f)(x)^2),x=-1..1);\n``=simplify(% );\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"x G-%%coshGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"x GF*-%%sinhGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF &-%$IntG6$*&-%%coshG6#%\"xGF&,&F&F&*$)-%%sinhGF.F%F&F&#F&F%/F/;!\"\"F& F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiGF(-%$I ntG6$*$)-%%coshG6#%\"xGF'F(/F2;!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiGF(,&*&-%%coshG6#F(F(-%%sinhGF.F (F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*Int(cosh*x*sqrt(1+sin^2*x),x = -1 .. 1) = 2*Pi*Int(cosh^2*x ,x = -1 .. 1);" "6#/*(\"\"#\"\"\"%#PiGF&-%$IntG6$*(%%coshGF&%\"xGF&-%% sqrtG6#,&F&F&*&%$sinGF%F-F&F&F&/F-;,$F&!\"\"F&F&*(F%F&F'F&-F)6$*&F,F%F -F&/F-;,$F&F7F&F&" }{XPPEDIT 18 0 "``= 2*Pi*(cosh*1*sinh*1+1)" "6#/%!G *(\"\"#\"\"\"%#PiGF',&**%%coshGF'F'F'%%sinhGF'F'F'F'F'F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 " _____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_________________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 66 "Find the are a of the surface formed when the section of the curve " }{XPPEDIT 18 0 "y = x^(3/2);" "6#/%\"yG)%\"xG*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 22 " from the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" } {TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\" \"" }{TEXT -1 22 " is rotated about the " }{TEXT 411 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 142 "f := x -> x^(3/2):\n'f'(x)=f(x);\nDiff('f'(x) ,x)=diff(f(x),x);\n2*Pi*Int(x*sqrt(1+D(f)(x)^2),x=0..1);\n``=simplify( %);\n``=simplify(value(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"fG6#%\"xG*$)F'#\"\"$\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%%DiffG6$-%\"fG6#%\"xGF*,$*(\"\"$\"\"\"\"\"#!\"\"F*#F.F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$,$*(F%!\"\"% \"xGF&,&\"\"%F&*&\"\"*F&F.F&F&#F&F%F&/F.;\"\"!F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&%#PiG\"\"\"-%$IntG6$*&%\"xGF',&\"\"%F'*&\"\"*F 'F,F'F'#F'\"\"#/F,;\"\"!F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$ **\"\"#\"\"\"\"%:7!\"\"%#PiGF(,&*&\"$Z#F(\"#8#F(F'F(\"#kF(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Pi*Int(x*sq rt(4+9*x),x = 0 .. 1)=2*Pi/1215" "6#/*&%#PiG\"\"\"-%$IntG6$*&%\"xGF&-% %sqrtG6#,&\"\"%F&*&\"\"*F&F+F&F&F&/F+;\"\"!F&F&*(\"\"#F&F%F&\"%:7!\"\" " }{XPPEDIT 18 0 " ``(64+247*sqrt(13))" "6#-%!G6#,&\"#k\"\"\"*&\"$Z#F( -%%sqrtG6#\"#8F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }} {PARA 0 "" 0 "" {TEXT -1 67 "Find the area of the surface formed when \+ the section of the curve " }{XPPEDIT 18 0 "y = 1/3;" "6#/%\"yG*&\"\" \"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(1/2) -x^(3/2)" "6#, &)%\"xG*&\"\"\"F'\"\"#!\"\"F')F%*&\"\"$F'F(F)F)" }{TEXT -1 22 " from t he point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x = 1/3;" "6#/%\"xG*&\"\"\"F&\" \"$!\"\"" }{TEXT -1 23 " is rotated about the " }{TEXT 412 1 "x" } {TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "f := x -> sqrt(x)/3-x^(3/2):\n'f'( x)=f(x);\nDiff('f'(x),x)=diff(f(x),x);\n2*Pi*Int(f(x)*sqrt(1+D(f)(x)^2 ),x=0..1/3);\n``=expand(simplify(%));\n``=simplify(value(rhs(%)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&\"\"$!\"\"F'#\"\"\" \"\"#F-*$)F'#F*F.F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-% \"fG6#%\"xGF*,&*&\"\"\"F-*&\"\"'F-F*#F-\"\"#!\"\"F-*(\"\"$F-F1F2F*#F-F 1F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$ *&,&*&\"\"$!\"\"%\"xG#F&F%F&*$)F0#F.F%F&F/F&,&F&F&*$),&*&F&F&*&\"\"'F& F0#F&F%F/F&*(F.F&F%F/F0F1F/F%F&F&F1/F0;\"\"!#F&F.F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"\"\"\"*F(*&%#PiGF(-%$IntG6$,(F(!\"\"* &\"\"'F(%\"xGF(F0*&\"#FF()F3\"\"#F(F(/F3;\"\"!#F(\"\"$F(F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"#F!\"\"%#PiG\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Pi*Int(``(1/9+2*x /3-3*x^2),x=0..1/3)=Pi/27" "6#/*&%#PiG\"\"\"-%$IntG6$-%!G6#,(*&F&F&\" \"*!\"\"F&*(\"\"#F&%\"xGF&\"\"$F0F&*&F4F&*$F3F2F&F0/F3;\"\"!*&F&F&F4F0 F&*&F%F&\"#FF0" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }} {PARA 0 "" 0 "" {TEXT -1 66 "Find the area of the surface formed when \+ the section of the curve " }{XPPEDIT 18 0 "y = x^2/4-ln*x/2;" "6#/%\"y G,&*&%\"xG\"\"#\"\"%!\"\"\"\"\"*(%#lnGF+F'F+F(F*F*" }{TEXT -1 22 " fro m the point where " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" } {TEXT -1 26 " is rotated (a) about the " }{TEXT 399 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 148 "f := x -> x^2/4-ln(x)/2:\n'f'(x)=f(x);\nDiff( 'f'(x),x)=diff(f(x),x);\n2*Pi*Int(x*sqrt(1+D(f)(x)^2),x=1..4);\n``=sim plify(%);\n``=simplify(value(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&\"\"%!\"\"F'\"\"#\"\"\"*&#F-F,F--%#lnGF&F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,&*&\"\"#! \"\"F*\"\"\"F/*&F/F/*&F-F/F*F/F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xGF&,&F&F&*$),&*&F%!\"\"F,F&F&*&F& F&*&F%F&F,F&F2F2F%F&F&#F&F%/F,;F&\"\"%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&%#PiG\"\"\"-%$IntG6$,&F'F'*$)%\"xG\"\"#F'F'/F.;F' \"\"%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"#C\"\"\"%#PiGF(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Pi*In t(``(1+x^2),x = 1 .. 4) = 24*Pi;" "6#/*&%#PiG\"\"\"-%$IntG6$-%!G6#,&F& F&*$%\"xG\"\"#F&/F/;F&\"\"%F&*&\"#CF&F%F&" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "f := x -> x^2/4-ln(x)/2:\n'f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x),x);\n2*Pi *Int(f(x)*sqrt(1+D(f)(x)^2),x=1..4);\n``=simplify(%);\n``=simplify(val ue(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&\"\" %!\"\"F'\"\"#\"\"\"*&#F-F,F--%#lnGF&F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,&*&\"\"#!\"\"F*\"\"\"F/*&F/F/*&F-F/F* F/F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$Int G6$*&,&*&\"\"%!\"\"%\"xGF%F&*&#F&F%F&-%#lnG6#F0F&F/F&,&F&F&*$),&*&F%F/ F0F&F&*&F&F&*&F%F&F0F&F/F/F%F&F&#F&F%/F0;F&F.F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"\"\"\"%F(*&%#PiGF(-%$IntG6$*(,&*$)%\"xG\" \"#F(F(*&F4F(-%#lnG6#F3F(!\"\"F(,&F1F(F(F(F(F3F9/F3;F(F)F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"\"\"#;F(*&%#PiGF(,(*&F)F( )-%#lnG6#\"\"#F2F(F(*&\"$G\"F(F/F(F(\"$:$!\"\"F(F(F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\" \"\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((-2*ln(x)+x^2)*(1+x^ 2)/x,x = 1 .. 4) =Pi/16" "6#/-%$IntG6$*(,&*&\"\"#\"\"\"-%#lnG6#%\"xGF+ !\"\"*$F/F*F+F+,&F+F+*$F/F*F+F+F/F0/F/;F+\"\"%*&%#PiGF+\"#;F0" } {XPPEDIT 18 0 "``(315-16*(ln*2)^2-128*ln*2);" "6#-%!G6#,(\"$:$\"\"\"*& \"#;F(*$*&%#lnGF(\"\"#F(F.F(!\"\"*(\"$G\"F(F-F(F.F(F/" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____ _________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 67 "Find the area of the surface formed when the section of the curve " }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$sinG\"\"\"%\"xG F'" }{TEXT -1 24 " from the point where " }{XPPEDIT 18 0 "x = 0;" "6 #/%\"xG\"\"!" }{TEXT -1 21 " to the point where " }{XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 23 " is rotated about the " }{TEXT 413 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "f := x -> sin(x):\n' f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x),x);\n2*Pi*Int(f(x)*sqrt(1+D(f)(x )^2),x=0..Pi);\n``=simplify(%);\n``=simplify(value(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*-%$cosGF)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&-%$sinG6#%\"xGF&,&F& F&*$)-%$cosGF.F%F&F&#F&F%/F/;\"\"!F'F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiGF(-%$IntG6$*&-%$sinG6#%\"xGF(,&F(F(*$)-% $cosGF0F'F(F(#F(F'/F1;\"\"!F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G,$*(\"\"#\"\"\"%#PiGF(,&*$F'#F(F'F(-%#lnG6#,&F(F(F+F(F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(sin(x)*sqrt(1+cos(x)^2),x);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$sinG6#%\"xG\"\"\",&F+F+*$)-%$cosGF)\"\"#F +F+#F+F1F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(*& -%$cosG6#%\"xGF(,&F(F(*$)F+F)F(F(#F(F)F(!\"\"*&#F(F)F(-%(arcsinhG6#F+F (F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "2*Pi*Int(sin*x*sqrt(1+cos^2*x),x = 0 .. Pi) =2*Pi*(s qrt(2)+ln(1+sqrt(2)))" "6#/*(\"\"#\"\"\"%#PiGF&-%$IntG6$*(%$sinGF&%\"x GF&-%%sqrtG6#,&F&F&*&%$cosGF%F-F&F&F&/F-;\"\"!F'F&*(F%F&F'F&,&-F/6#F%F &-%#lnG6#,&F&F&-F/6#F%F&F&F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Note:" }{XPPEDIT 18 0 "Int(sin*x*sqrt(1+cos^2*x),x)=-1/2" "6#/-%$IntG6$*(%$sinG\"\"\"%\"xGF)-%%sqrtG6#,&F)F)*&%$cosG\"\"#F*F)F)F )F*,$*&F)F)F1!\"\"F4" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x*sqrt(1+cos ^2*x)-1/2;" "6#,&*(%$cosG\"\"\"%\"xGF&-%%sqrtG6#,&F&F&*&F%\"\"#F'F&F&F &F&*&F&F&F-!\"\"F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsinh(cos*x))+c " "6#,&-%(arcsinhG6#*&%$cosG\"\"\"%\"xGF)F)%\"cGF)" }{TEXT -1 3 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 37 "pictur es of cylinder and cone etc. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1990 "p1 := plot([cos(t),0.3*sin (t),t=-Pi..0],color=navy):\np2 := plot([cos(t),0.3*sin(t)+2,t=0..2*Pi] ,color=navy):\np3 := plot([[[1,0],[1,2]],[[-1,0],[-1,2]]],color=navy): \na := evalf(Pi/60):\nc1 := seq([cos(j*a),0.3*sin(j*a)],j=-12..0):\nc2 := seq([cos(j*a),0.3*sin(j*a)],j=-27..-12):\nc3 := seq([cos(j*a),0.3* sin(j*a)],j=-33..-27):\nc4 := seq([cos(j*a),0.3*sin(j*a)],j=-48..-33): \nc5 := seq([cos(j*a),0.3*sin(j*a)],j=-60..-48):\nc6 := seq([cos(j*a), 2-0.3*sin(j*a)],j=0..12):\nc7 := seq([cos(j*a),2-0.3*sin(j*a)],j=12..2 7):\nc8 := seq([cos(j*a),2-0.3*sin(j*a)],j=27..33):\nc9 := seq([cos(j* a),2-0.3*sin(j*a)],j=33..48):\nc10 := seq([cos(j*a),2-0.3*sin(j*a)],j= 48..60):\np4 := plots[polygonplot]([c1,c6],\n style=patchnogrid,col or=COLOR(RGB,.95,.90,1)):\np5 := plots[polygonplot]([c2,c7],\n styl e=patchnogrid,color=COLOR(RGB,.95,.91,1)):\np6 := plots[polygonplot]([ c3,c8],\n style=patchnogrid,color=COLOR(RGB,.953,.923,1)):\np7 := p lots[polygonplot]([c4,c9],\n style=patchnogrid,color=COLOR(RGB,.95, .91,1)):\np8 := plots[polygonplot]([c5,c10],\n style=patchnogrid,co lor=COLOR(RGB,.95,.90,1)):\nc11 := seq([cos(2*j*a),0.3*sin(2*j*a)+2],j =0..60):\np9 := plots[polygonplot]([c11],\n style=patchnogrid,color =COLOR(RGB,.953,.933,1)):\np10 := plot([[[1,0],[1.2,0]],[[1,2],[1.2,2] ],\n [[0,-.3],[0,1.7]]],color=navy,linestyle=[2,2,3]):\np11 : = plottools[arrow]([1.15,.87],[1.15,0],\n 0,.08,.12,arrow,col or=COLOR(RGB,.35,.5,.75)):\np12 := plottools[arrow]([1.15,1.13],[1.15, 2],\n 0,.08,.12,arrow,color=COLOR(RGB,.35,.5,.75)):\np13 := p lottools[arrow]([.6,2],[1,2],\n 0,.1,.24,arrow,color=COLOR(RG B,.35,.5,.75)):\np14 := plot([[0,2],[.4,2]],color=COLOR(RGB,.35,.5,.75 )):\np15 := plot([[[0,2]]$3],style=point,\n symbol=[cross,diamond, circle],color=COLOR(RGB,.35,.5,.75)):\nt1 := plots[textplot]([[1.15,1, `h`],[.5,2,`r`]],\n font=[HELVETICA,9],color=COLOR (RGB,.35,.5,.75)):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,\n \+ p10,p11,p12,p13,p14,p15,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 908 "p1 := plot([[0,0],[0,1],[2,1],[2,0],[0,0]],color= navy):\np2 := plots[polygonplot]([[0,0],[0,1],[2,1],[2,0]],\n style =patchnogrid,color=COLOR(RGB,.953,.933,1)):\np3 := plot([[[-.1,0],[0,0 ]],[[-.1,1],[0,1]],\n [[0,0],[0,-.15]],[[2,0],[2,-.15]]],colo r=navy,linestyle=2):\np4 := plottools[arrow]([-.07,.6],[-.07,1],\n \+ 0,.04,.12,arrow,color=COLOR(RGB,.35,.5,.75)):\np5 := plottools[a rrow]([-.07,.4],[-.07,0],\n 0,.04,.12,arrow,color=COLOR(RGB,. 35,.5,.75)):\np6 := plottools[arrow]([1.07,-.1],[2,-.1],\n 0, .06,.03,arrow,color=COLOR(RGB,.35,.5,.75)):\np7 := plottools[arrow]([. 93,-.1],[0,-.1],\n 0,.06,.03,arrow,color=COLOR(RGB,.35,.5,.75 )):\nt1 := plots[textplot]([[-0.06,.5,'`h`'],[1,-.1,`2 r`]],\n \+ font=[HELVETICA,9],color=navy):\nt2 := plots[te xtplot]([1,-.1,`p`],font=[SYMBOL,9],color=navy):\nplots[display]([p1,p 2,p3,p4,p5,p6,p7,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1465 "p1 := plot([cos(t),0.3*s in(t),t=-Pi..0],color=navy):\np2 := plot([cos(t),0.3*sin(t),t=0..Pi], \n color=COLOR(RGB,.7,.7,.8),linestyle=2):\np3 := p lot([[[1,0],[0,2]],[[-1,0],[0,2]]],color=navy):\na := evalf(Pi/60):\nc 1 := seq([cos(j*a),0.3*sin(j*a)],j=-12..0):\nc2 := seq([cos(j*a),0.3*s in(j*a)],j=-27..-12):\nc3 := seq([cos(j*a),0.3*sin(j*a)],j=-33..-27): \nc4 := seq([cos(j*a),0.3*sin(j*a)],j=-48..-33):\nc5 := seq([cos(j*a), 0.3*sin(j*a)],j=-60..-48):\np4 := plots[polygonplot]([c1,[0,2]],\n \+ style=patchnogrid,color=COLOR(RGB,.95,.90,1)):\np5 := plots[polygonplo t]([c2,[0,2]],\n style=patchnogrid,color=COLOR(RGB,.95,.91,1)):\np6 := plots[polygonplot]([c3,[0,2]],\n style=patchnogrid,color=COLOR( RGB,.953,.923,1)):\np7 := plots[polygonplot]([c4,[0,2]],\n style=pa tchnogrid,color=COLOR(RGB,.95,.91,1)):\np8 := plots[polygonplot]([c5,[ 0,2]],\n style=patchnogrid,color=COLOR(RGB,.95,.90,1)):\nc11 := seq ([cos(2*j*a),0.3*sin(2*j*a)+2],j=0..60):\np9 := plot([[[cos(-40*a),0.3 *sin(-40*a)],[0,2]]],color=navy,linestyle=[2,3]):\np10 := plottools[ar row]([.6,0],[1,0],\n 0,.1,.24,arrow,color=COLOR(RGB,.35,.5,.7 5)):\np11 := plot([[0,0],[.4,0]],color=COLOR(RGB,.35,.5,.75)):\np12 := plot([[[0,0]]$3],style=point,\n symbol=[cross,diamond,circle],col or=COLOR(RGB,.35,.5,.75)):\nt1 := plots[textplot]([[.63,1.03,`d`],[.5, 0,`r`]],\n font=[HELVETICA,9],color=navy): \nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,t1],axes=none );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 511 "a := evalf(Pi/60):\nc1 := seq([cos(j*a),sin(j*a)],j= -4..50):\np1 := plot([[0,0],c1,[0,0]],color=navy):\np2 := plots[polygo nplot]([[0,0],c1,[0,0]],\n style=patchnogrid,color=COLOR(RGB,.953,. 933,1)):\nc2 := seq([.21*cos(2*j*a),.21*sin(2*j*a)],j=-2..25):\np3 := \+ plot([c2],color=COLOR(RGB,.6,.6,.6)):\nt1 := plots[textplot]([.03,.09, `q`],font=[SYMBOL,10],color=navy):\nt2 := plots[textplot]([[-.43,.16,` d`],[.45,-.18,`d`]],font=[HELVETICA,9],color=navy):\nplots[display]([p 1,p2,p3,t1,t2],axes=none,scaling=unconstrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2166 "p1 := plot ([cos(t),0.3*sin(t),t=-Pi..0],color=navy):\np2 := plot([.5*cos(t),0.15 *sin(t)+2,t=0..2*Pi],color=navy):\np3 := plot([[[1,0],[.5,2]],[[-1,0], [-.5,2]]],color=navy):\na := evalf(Pi/60):\nc1 := seq([cos(j*a),0.3*si n(j*a)],j=-12..0):\nc2 := seq([cos(j*a),0.3*sin(j*a)],j=-27..-12):\nc3 := seq([cos(j*a),0.3*sin(j*a)],j=-33..-27):\nc4 := seq([cos(j*a),0.3* sin(j*a)],j=-48..-33):\nc5 := seq([cos(j*a),0.3*sin(j*a)],j=-60..-48): \nc6 := seq([.5*cos(j*a),2-0.15*sin(j*a)],j=0..12):\nc7 := seq([.5*cos (j*a),2-0.15*sin(j*a)],j=12..27):\nc8 := seq([.5*cos(j*a),2-0.15*sin(j *a)],j=27..33):\nc9 := seq([.5*cos(j*a),2-0.15*sin(j*a)],j=33..48):\nc 10 := seq([.5*cos(j*a),2-0.15*sin(j*a)],j=48..60):\np4 := plots[polygo nplot]([c1,c6],\n style=patchnogrid,color=COLOR(RGB,.95,.90,1)):\np 5 := plots[polygonplot]([c2,c7],\n style=patchnogrid,color=COLOR(RG B,.95,.91,1)):\np6 := plots[polygonplot]([c3,c8],\n style=patchnogr id,color=COLOR(RGB,.953,.923,1)):\np7 := plots[polygonplot]([c4,c9],\n style=patchnogrid,color=COLOR(RGB,.95,.91,1)):\np8 := plots[polygo nplot]([c5,c10],\n style=patchnogrid,color=COLOR(RGB,.95,.90,1)):\n c11 := seq([.5*cos(2*j*a),0.15*sin(2*j*a)+2],j=0..60):\np9 := plots[po lygonplot]([c11],\n style=patchnogrid,color=COLOR(RGB,.953,.933,1)) :\np10 := plottools[arrow]([.3,2],[.5,2],\n 0,.05,.23,arrow,c olor=COLOR(RGB,.35,.5,.75)):\np11 := plottools[arrow]([.57,0],[1,0],\n 0,.08,.12,arrow,color=COLOR(RGB,.35,.5,.75)):\np12 := plot([ [[0,2],[.2,2]],[[0,0],[.45,0]]],color=COLOR(RGB,.35,.5,.75)):\np13 := \+ plot([[[0,0],[0,2]]$3],style=point,\n symbol=[cross,diamond,circle ],color=COLOR(RGB,.35,.5,.75)):\np14 := plot([cos(t),0.3*sin(t),t=0..P i],\n color=COLOR(RGB,.35,.5,.75),linestyle=2):\np15 := plot([[[ -.5,2],[0,4]],[[.5,2],[0,4]],[[0,2],[0,4]]],\n color=navy,lines tyle=2):\nt1 := plots[textplot]([[.24,2.02,`r`],[.5,.01,`r`],\n \+ [.8,1.14,`d`],[.3,3.06,`d`]],\n font=[HELVE TICA,9],color=navy):\nt2 := plots[textplot]([[.27,1.96,`1`],[.54,-.03, `2`],[.34,3,`1`]],\n font=[HELVETICA,8],color=navy): \nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,\n p10,p11,p12, p13,p14,p15,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 23 "surface area pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2047 "f := x -> 1.1-0.6*cos( x+0.1):\na := .5: b := 2.5: c := 1.6: d := 1.66:\nfa := f(a):fb := f(b ):fc := f(c):fd := f(d):\np1 := plot([f(x),-f(x)],x=0..2.8,color=red,l inestyle=[1,2]):\np2 := plot([f(x),-f(x)],x=a..d,color=blue,thickness= 2,linestyle=[1,2]):\np3 := plot([[[0,0],[3,0]],[[a,0],[a,fa]],[[b,0],[ b,fb]]],\n color=COLOR(RGB,.3,.3,.3),linest yle=[1,3,3]):\np4 := plot([[[c,0],[c,fc]],[[d,0],[d,fd]],\n [[( c+d)/2,0],[(c+d)/2,-.1]]],color=black,linestyle=2):\np5 := plottools[a rrow]([c-.19,.2],[c,.2],0,.06,.3,arrow,color=black):\np6 := plottools[ arrow]([d+.19,.2],[d,.2],0,.06,.3,arrow,color=black):\np7 := plots[pol ygonplot]([[c,f(c)],[d,f(d)],[d,f(c)]],color=blue):\np8 := plot([a+.14 *fa*cos(t),fa*sin(t),t=0..2*Pi],color=COLOR(RGB,.3,0,.7)):\np9 := plot ([b+.14*fb*cos(t),fb*sin(t),t=0..2*Pi],color=COLOR(RGB,.3,0,.7)):\np10 := plot([c+.14*fc*cos(t),fc*sin(t),t=0..2*Pi],\n \+ color=COLOR(RGB,.3,0,.7),linestyle=2):\np11 := plot([d+.14*fd*cos(t ),fd*sin(t),t=0..2*Pi],\n color=COLOR(RGB,.3,0, .7),linestyle=2):\nalpha := evalf(Pi/30): h := (d-a)/15:\nc1 := seq([a +.14*fa*cos(j*alpha),fa*sin(j*alpha)],j=30..45):\nc2 := seq([a+j*h,-f( a+j*h)],j=0..15):\nc3 := seq([d+.14*fd*cos(j*alpha),fd*sin(j*alpha)],j =-15..15):\nc4 := seq([d-j*h,f(d-j*h)],j=0..15):\nc5 := seq([a+.14*fa* cos(j*alpha),fa*sin(j*alpha)],j=15..30):\np12 := plots[polygonplot]([c 1,c2,c3,c4,c5],\n style=patchnogrid,color=COLOR(RGB,.98,.94 ,1)):\nt1 := plots[textplot]([[1.57,.305,'d']],font=[SYMBOL,11],color= black):\nt2 := plots[textplot]([[1.62,.3,`x`],[.5,-.07,`x=a`],\n \+ [2.5,-.07,`x=b`],[3,-.05,`x`],[.43,.75,`(a,f(a))`],\n [2.47,1.75, `(b,f(b))`],[(c+d)/2,-.15,`x`]],font=[HELVETICA,10],color=black):\nt3 \+ := plots[textplot]([1.97,1.59,`y = f(x)`],font=[HELVETICA,10],color=re d):\nt4 := plots[textplot]([[1.57,1.295,'d']],font=[SYMBOL,11],color=b lue):\nt5 := plots[textplot]([[.97,.94,`s`],[1.62,1.29,`s`]],font=[HEL VETICA,10],color=blue):\nplots[display]([p||(1..12),t||(1..5)],axes=no ne,view=[0..3,-1.75..1.76]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 564 "interp([0,.5,1],[0,.29,.55],x):\np := unapply(%,x):\np1 := plot(p (x),x=-.2..1.2):\np2 := plot([[0,0],[1,0],[1,.55],[0,0]],color=black): \np3 := plots[polygonplot]([[0,0],[1,0],[1,.55]],color=COLOR(RGB,.6,.6 ,1)):\nt1 := plots[textplot]([[.52,-.048,'d'],[1.05,.3,'d']],font=[SYM BOL,11],color=black):\nt2 := plots[textplot]([.45,.342,'d'],font=[SYMB OL,11],color=blue):\nt3 := plots[textplot]([[.56,-.05,`x`],[1.09,.3,`y `]],font=[HELVETICA,10],color=black):\nt4 := plots[textplot]([.49,.34, `s`],font=[HELVETICA,10],color=blue):\nplots[display]([p1,p2,p3,t1,t2, t3,t4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 904 "interp([0,.5,1],[0,.35,.55],x):\np := unap ply(%,x):\np1 := plot(p(x),x=-.2..1.3):\np2 := plot([[0,0],[1,0],[1,.5 5],[0,0]],color=black):\np3 := plot([[[0,0],[.5,.35],[1,.55]]$3],style =point,\n symbol=[circle,diamond,cross],color=black):\n p4 := plot([[.1,.13],[.9,.57]],color=blue):\np5 := plot([[[0,-.15],[0, 0]],[[1,-.15],[1,0]],[[.5,-.15],[.5,.35]]],\n col or=black,linestyle=2):\nt1 := plots[textplot]([[-.08,.05,`P`],[.96,.6, `P`],[.44,.39,`P`],[1.045,0,`Q`],\n [.47,.4,`*`],[-.01,-.19,`x`],[.49 ,-.19,`x`],[.99,-.19,`x`],[.51,-.18,`*`]],\n font=[HELVETICA,10],col or=black):\nt2 := plots[textplot]([[-.04,.03,`i-1`],[.98,.58,`i`],[.46 ,.37,`i`],\n [.04,-.21,`i-1`],[.52,-.21,`i`],[1.02,-.21,`i`]] ,\n font=[HELVETICA,9],color=black):\nt3 := plots[textp lot]([1.25,.66,`y = f(x)`],font=[HELVETICA,10],color=red):\nplots[disp lay]([p||(1..5),t||(1..3)],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2081 "f := x -> 1.1-0.6*cos (x+0.1):\na := .5: b := 2.5: c := 1.6: d := 1.66:\nfa := f(a):fb := f( b):fc := f(c):fd := f(d):\np1 := plot(f(x),x=0.1..2.8,color=red):\np2 \+ := plot(f(x),x=a..d,color=blue,thickness=2):\np3 := plot(f(-x),x=-2.7. .-0.1,color=red,linestyle=2):\np4 := plot(f(-x),x=-d..-a,color=blue,th ickness=2,linestyle=2):\np5 := plot([[[0,-.2],[0,2.1]],[[-3,0],[3,0]], [[a,0],[a,fa]],[[b,0],[b,fb]]],\n color=COL OR(RGB,.3,.3,.3),linestyle=[1$2,3$2]):\np6 := plot([[[c,0],[c,fc]],[[d ,0],[d,fd]],\n [[(c+d)/2,0],[(c+d)/2,-.1]]],color=black,linesty le=2):\np7 := plottools[arrow]([c-.19,.2],[c,.2],0,.06,.3,arrow,color= black):\np8 := plottools[arrow]([d+.19,.2],[d,.2],0,.06,.3,arrow,color =black):\np9 := plots[polygonplot]([[c,f(c)],[d,f(d)],[d,f(c)]],color= blue):\np10 := plot([a*cos(t),fa+.14*a*sin(t),t=0..2*Pi],color=COLOR(R GB,.3,0,.7)):\np11 := plot([b*cos(t),fb+.14*b*sin(t),t=0..2*Pi],color= COLOR(RGB,.3,0,.7)):\np12 := plot([c*cos(t),fc+.14*c*sin(t),t=0..2*Pi] ,\n color=COLOR(RGB,.3,0,.7),linestyle=2):\np13 := plot([d*cos(t),fd+.14*d*sin(t),t=0..2*Pi],\n \+ color=COLOR(RGB,.3,0,.7),linestyle=2):\nalpha := evalf(Pi/30): h := \+ (d-a)/15:\nc1 := seq([a*cos(j*alpha),fa+.14*a*sin(j*alpha)],j=-15..0): \nc2 := seq([a+j*h,f(a+j*h)],j=0..15):\nc3 := seq([d*cos(j*alpha),fd+. 14*d*sin(j*alpha)],j=0..30):\nc4 := seq([j*h-d,f(d-j*h)],j=0..15):\nc5 := seq([a*cos(j*alpha),fa+.14*a*sin(j*alpha)],j=30..45):\np14 := plot s[polygonplot]([c1,c2,c3,c4,c5],\n style=patchnogrid,color= COLOR(RGB,.98,.94,1)):\nt1 := plots[textplot]([[-0.05,2.1,`y`],[1.49,. 27,`x`],[.5,-.07,`x = a`],\n [2.5,-.07,`x = b`],[3,-.05,`x`],[.75 ,.55,`(a,f(a))`],\n [2.75,1.55,`(b,f(b))`],[(c+d)/2,-.15,`x`]],fo nt=[HELVETICA,10],color=black):\nt2 := plots[textplot]([1.97,1.59,`y = f(x)`],font=[HELVETICA,10],color=red):\nt3 := plots[textplot]([[1.57, 1.295,'d']],font=[SYMBOL,11],color=blue):\nt4 := plots[textplot]([[.97 ,.94,`s`],[1.62,1.29,`s`]],font=[HELVETICA,10],color=blue):\nplots[dis play]([p||(1..14),t||(1..4)],axes=none,view=[-2.7..3,-.2..2.1]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "summary pictures " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2110 "f := x -> 1.1-0.6*cos(x+0.5)+0.2*sin(3*x+0.9):\na:=1.:fa:=f(a):b :=1.7:fb:=f(b):c:=2.7:fc:=f(c):d:=4.:fd:=f(d):e:=5.:fe:=f(e):\np1 := p lot([f(x),-f(x)],x=1..5,y=0..2,color=red,thickness=[2,1]):\nalpha := e valf(Pi/16):h1 := (b-a)/10:\nh2 := (c-b)/10:h3 := (d-c)/10:h4 := (e-d) /10:\nc1 := seq([a+.14*fa*cos(j*alpha),fa*sin(j*alpha)],j=-8..8):\nc2 \+ := seq([a+.14*fa*cos(j*alpha),fa*sin(j*alpha)],j=8..24):\nc3 := seq([b +.14*fb*cos(j*alpha),fb*sin(j*alpha)],j=-8..8):\nc4 := seq([c+.14*fc*c os(j*alpha),fc*sin(j*alpha)],j=-8..8):\nc5 := seq([d+.14*fd*cos(j*alph a),fd*sin(j*alpha)],j=-8..8):\nc6 := seq([e+.14*fe*cos(j*alpha),fe*sin (j*alpha)],j=-8..8):\nc7 := seq([e+.14*fe*cos(j*alpha),fe*sin(j*alpha) ],j=8..24):\nc8 := seq([a+j*h1,-f(a+j*h1)],j=0..10):\nc9 := seq([b+j*h 2,-f(b+j*h2)],j=0..10):\nc10 := seq([c+j*h3,-f(c+j*h3)],j=0..10):\nc11 := seq([d+j*h4,-f(d+j*h4)],j=0..10):\nc12 := seq([e-j*h4,f(e-j*h4)],j =0..10):\nc13 := seq([d-j*h3,f(d-j*h3)],j=0..10):\nc14 := seq([c-j*h2, f(c-j*h2)],j=0..10):\nc15 := seq([b-j*h1,f(b-j*h1)],j=0..10):\np2 := p lot([c1,c2],color=COLOR(RGB,.3,0,.7)):\np3 := plot([c6],color=COLOR(RG B,.3,0,.7)):\np4 := plot([c7],color=COLOR(RGB,.3,0,.7),linestyle=2):\n p5 := plots[polygonplot]([[1.3,0],c8,c3,c15,c2],style=patchnogrid,\n \+ color=COLOR(RGB,.975,.935,1)):\np6 := plots[polygonp lot]([[2.,0],c9,c4,c14,c3],style=patchnogrid,\n col or=COLOR(RGB,.99,.95,1)):\np7 := plots[polygonplot]([[3.5,0],c10,c5,c1 3,c4],style=patchnogrid,\n color=COLOR(RGB,.975,.93 5,1)):\np8 := plots[polygonplot]([[4.5,0],c11,c6,c12,c5],style=patchno grid,\n color=COLOR(RGB,.96,.92,1)):\np9 := plot([[ .5,0],[5.5,0]],color=black):\np10 := plot(f(x),x=0.5..5.3,y=0..2,color =red,thickness=1):\np11 := plot([[[a,f(a)],[e,f(e)]]$3],style=point,sy mbol=[circle,diamond,cross],\n color=black):\nt1 \+ := plots[textplot]([[5.5,-.11,`x`],[.8,1.28,`(a,f(a))`],\n [5.25, .95,`(b,f(b))`]],font=[HELVETICA,10],color=black):\nt2 := plots[textpl ot]([3.4,1.82,`y = f(x)`],font=[HELVETICA,10],color=red):\nplots[displ ay]([p||(1..11),t1,t2],view=[0..5.5,-2..2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1973 "f := \+ x -> 1.7+.8*x-1.8*sin(x+0.1)+.3*cos(4*x+0.3):\na:=1.2:fa:=f(a):b:=2.:f b:=f(b):c:=3.3:fc:=f(c):d:=4.:fd:=f(d):\np1 := plot(f(x),x=a..d,color= red,thickness=2):\np2 := plot(f(-x),x=-d..-a,color=red,thickness=1):\n alpha := evalf(Pi/16):\nh1 := (b-a)/10:h2 := (c-b)/10:h3 := (d-c)/10: \nc1 := seq([a*cos(j*alpha),fa+.14*a*sin(j*alpha)],j=0..16):\nc2 := se q([a*cos((-j)*alpha),fa+.14*a*sin((-j)*alpha)],j=0..16):\nc3 := seq([b *cos(j*alpha),fb+.14*b*sin(j*alpha)],j=16..32):\nc4 := seq([b*cos((-j) *alpha),fb+.14*b*sin((-j)*alpha)],j=0..16):\nc5 := seq([c*cos(j*alpha) ,fc+.14*c*sin(j*alpha)],j=16..32):\nc6 := seq([c*cos((-j)*alpha),fc+.1 4*c*sin((-j)*alpha)],j=0..16):\nc7 := seq([d*cos(j*alpha),fd+.14*d*sin (j*alpha)],j=0..16):\nc8 := seq([d*cos(j*alpha),fd+.14*d*sin(j*alpha)] ,j=16..32):\nc9 := seq([a+j*h1,f(a+j*h1)],j=0..10):\nc10 := seq([b+j*h 2,f(b+j*h2)],j=0..10):\nc11 := seq([c+j*h3,f(c+j*h3)],j=0..10):\nc12 : = seq([j*h1-b,f(b-j*h1)],j=0..10):\nc13 := seq([j*h2-c,f(c-j*h2)],j=0. .10):\nc14 := seq([j*h3-d,f(d-j*h3)],j=0..10):\np3 := plots[polygonplo t]([[0,f(1.5)],c9,c4,c12,c2],style=patchnogrid,\n c olor=COLOR(RGB,.965,.925,1)):\np4 := plots[polygonplot]([[0,f(2.5)],c1 0,c6,c13,c4],style=patchnogrid,\n color=COLOR(RGB,. 99,.95,1)):\np5 := plots[polygonplot]([[0,f(3.5)],c11,c7,c14,c6],style =patchnogrid,\n color=COLOR(RGB,.975,.935,1)):\np6 \+ := plot([c1],color=COLOR(RGB,.3,0,.7),linestyle=2):\np7 := plot([c2],c olor=COLOR(RGB,.3,0,.7)):\np8 := plot([c7,c8],color=COLOR(RGB,.3,0,.7) ):\np9 := plot(f(x),x=.9..4.3,color=red,thickness=1):\np10 := plot([[[ a,f(a)],[d,f(d)]]$3],style=point,symbol=[circle,diamond,cross],\n \+ color=black):\nt1 := plots[textplot]([[4.5,-.15,`x`], [-.13,7.8,`y`],[1.67,.84,`(a,f(a))`],\n [4.45,5.9,`(b,f(b))`]],fo nt=[HELVETICA,10],color=black):\nt2 := plots[textplot]([3.2,3,`y = f(x )`],font=[HELVETICA,10],color=red):\nplots[display]([p||(1..10),t1,t2] ,view=[-4.5..4.5,-.3..7.8],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }