(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 334664, 9302]*) (*NotebookOutlinePosition[ 335357, 9327]*) (* CellTagsIndexPosition[ 335313, 9323]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Examples for Section 9.5 Calculus with Parameterized Curves\ \>", "Subsection"], Cell[CellGroupData[{ Cell["p. 749", "Subsection"], Cell[CellGroupData[{ Cell["3.", "Subsubsection"], Cell[TextData[{ "Find the equation of the line tangent to ", Cell[BoxData[ \(TraditionalForm\`x\ = \ 4\ \(sin(t)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y\ = \ 2\ \(cos(t)\)\)]], " at the point corresponding to ", Cell[BoxData[ \(TraditionalForm\`t = \[Pi]/4\)]], ".\n\n", StyleBox["Solution. 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\(TraditionalForm\`x\ = \ cos(t)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y\ = \ t\ + \ sin(t)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`0 \[LessEqual] t \[LessEqual] \[Pi]\)]], ".\n\n", StyleBox["Solution. ", FontSlant->"Italic"], "The length of the curve is ", Cell[BoxData[ \(TraditionalForm\`L = 4\)]], ", by the preceding problem 17. The coordinates of the centroid are the \ weighted averages of ", Cell[BoxData[ \(TraditionalForm\`x\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\)]], ":", StyleBox["\n", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(x\&_ = \), "TraditionalForm"], \(1\/L\), \(\[Integral]\_0\%\[Pi] x \(\@\(\((\(d\ x\)\/\(d\ t\))\)\^2 + \((\(d\ y\)\/\(d\ t\))\)\^2\)\) d\ t\)}], "=", " ", \(\(1\/4\) \(\[Integral]\_0\%\[Pi]\ \(cos(t)\) \(\@\(2 + 2 \( cos(t)\)\)\) d\ t\) = \(\@2\/4\) \(\[Integral]\_0\%\[Pi]\ \(\(cos(t)\) \(sin(t)\)\)\/\@\(1 - cos(t)\)\ d\ t\)\)}], TraditionalForm]]], "\nSubstitute ", Cell[BoxData[ \(TraditionalForm\`u = 1 - cos(t)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`d\ u\ = \ \(sin(t)\) d\ t\)]], ", ", Cell[BoxData[ \(TraditionalForm\`cos(t)\ = \ 1 - u\)]], ":\n", Cell[BoxData[ \(TraditionalForm \`x\&_ = \(\(\@2\/4\) \(\[Integral]\_0\%2\ \((1 - u)\)\/\@\(u)\)\ d\ u\) = \(\(\(\@2\/4\) \((2 u\^\(1/2\) - \(2\/3\) u\^\(3/2\))\)\( | \_0\%2\)\) = \(1 - 2\/3 = 1\/3\)\)\)\)]], ".\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(y\&_ = \), "TraditionalForm"], \(1\/L\), \(\[Integral]\_0\%\[Pi] y \(\@\(\((\(d\ x\)\/\(d\ t\))\)\^2 + \((\(d\ y\)\/\(d\ t\))\)\^2\)\) d\ t\)}], "=", " ", \(\(1\/4\) \(\[Integral]\_0\%\[Pi]\((t + sin(t))\) \(\@\(2 + 2 \( cos(t)\)\)\) d\ t\) = \ \(\@2\/4\) \(\[Integral]\_0\%\[Pi] t \(\@\( 1 + cos(t)\)\) d\ t\) + \ \(\@2\/4\) \(\[Integral]\_0\%\[Pi]\( sin(t)\) \(\@\(1 + cos(t)\)\) d\ t \)\)}], TraditionalForm]]], "\nThe first integral can be done with integration by parts (", Cell[BoxData[ \(TraditionalForm\`u = t\)]], ", ", Cell[BoxData[ \(TraditionalForm\`d\ v\ = \ \@\(1 + cos(t)\)\)]], "; use the fact discovered in the previous problem that ", Cell[BoxData[ \(TraditionalForm \`v = \(\[Integral]\@\(1 + cos(t)\)\ d\ t = \ 2 \@\( 1 - cos(t)\)\)\)]], "), and the second is an easy substitution. The final answer is \n", Cell[BoxData[ \(TraditionalForm \`y\&_ = \(\(\(\@2\/4\) \((2 t \@\( 1 - cos(t)\)\ + \ 4 \@\( 1 + cos(t)\) - \(2\/3\) \((\@\(1 + cos(t)\))\)\^\(3/2\))\)\( | \_0 \%\[Pi]\)\) = \ \[Pi] - 4\/3\)\)]], ". \nIt is much easier to let ", StyleBox["Mathematica", FontSlant->"Italic"], " take care of the details:" }], "Text"], Cell[BoxData[ \(x[t_] := \ Cos[t]; \ y[t_]\ := \ t + Sin[t]; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(L = \[Integral]\_0\%\[Pi]\(\@\(\( x'\)[t]\^2 + \(y'\)[t]\^2\)\) \[DifferentialD]t\)], "Input"], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(xbar = \(1\/L\) \(\[Integral]\_0\%\[Pi] x[t] \(\@\(\(x'\)[t]\^2 + \(y'\)[t]\^2\)\) \[DifferentialD]t\)\)], "Input"], Cell[BoxData[ \(1\/3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ybar\ = \ \(1\/L\) \(\[Integral]\_0\%\[Pi] y[t] \(\@\(\(x'\)[t]\^2 + \(y'\)[t]\^2\)\) \[DifferentialD]t\)\ // Simplify\)], "Input"], Cell[BoxData[ \(\(-\(4\/3\)\) + \[Pi]\)], "Output"] }, Open ]], Cell[TextData[{ "From the previous problem we have the plot of the curve stored in ", StyleBox["p", "Input"], ". 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1.22346 m .75195 1.19594 L .7579 1.09014 L p 0 0 .303 r F P 0 g s .76845 1.06705 m .7579 1.09014 L .75195 1.19594 L p 0 0 .235 r F P 0 g s .62762 1.40508 m .64217 1.39377 L .67725 1.29674 L p .147 0 .16 r F P 0 g s .69226 1.28376 m .67725 1.29674 L .64217 1.39377 L p .045 0 .113 r F P 0 g s .46644 .47911 m .49016 .48573 L .41375 .42338 L p .155 .561 .93 r F P 0 g s .41375 .42338 m .38646 .41429 L .46644 .47911 L p .155 .561 .93 r F P 0 g s .69692 .71924 m .70963 .72676 L .66586 .64172 L p 0 .106 .66 r F P 0 g s .66586 .64172 m .64766 .62374 L .69692 .71924 L p 0 .106 .66 r F P 0 g s .64217 1.39377 m .65836 1.37854 L .69226 1.28376 L p .045 0 .113 r F P 0 g s .70795 1.26732 m .69226 1.28376 L .65836 1.37854 L p 0 0 .058 r F P 0 g s .76845 1.06705 m .77616 1.04164 L .76403 .93737 L p 0 0 .433 r F P 0 g s .76954 .92141 m .76403 .93737 L .77616 1.04164 L p 0 0 .341 r F P 0 g s .76403 .93737 m .76954 .92141 L .74518 .82277 L p 0 0 .52 r F P 0 g s .62774 .60925 m .64766 .62374 L .59299 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.28948 .40959 m .29152 .40894 L .2064 .3903 L p .547 .896 .978 r F P 0 g s .2064 .3903 m .20427 .39098 L .28948 .40959 L p .547 .896 .978 r F P 0 g s .29152 .40894 m .29768 .40749 L .2129 .3888 L p .471 .864 .971 r F P 0 g s .2129 .3888 m .2064 .3903 L .29152 .40894 L p .471 .864 .971 r F P 0 g s .75121 .81767 m .75017 .81967 L .77124 .90492 L p 0 0 .234 r F P 0 g s .75017 .81967 m .75121 .81767 L .72581 .74901 L p 0 0 .324 r F P 0 g s .72581 .74901 m .71944 .73665 L .75017 .81967 L p 0 0 .324 r F P 0 g s .29768 .40749 m .30795 .40598 L .22389 .38729 L p .373 .815 .952 r F P 0 g s .22389 .38729 m .2129 .3888 L .29768 .40749 L p .373 .815 .952 r F P 0 g s .70965 1.30647 m .72517 1.27325 L .75195 1.19594 L p 0 0 0 r F P 0 g s .76317 1.16477 m .75195 1.19594 L .72517 1.27325 L p 0 0 0 r F P 0 g s .66586 .64172 m .68174 .66332 L .64036 .60672 L p 0 0 .421 r F P 0 g s .64036 .60672 m .61755 .57564 L .66586 .64172 L p 0 0 .421 r F P 0 g s .30795 .40598 m .32221 .40518 L .23935 .38659 L p 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.25918 .3875 L .34027 .40582 L p 0 .555 .784 r F P 0 g s .72517 1.27325 m .73867 1.23552 L .76317 1.16477 L p 0 0 0 r F P 0 g s .77151 1.13013 m .76317 1.16477 L .73867 1.23552 L p .776 .786 .288 r F P 0 g s .60013 1.42016 m .59807 1.42082 L .53436 1.4749 L p .108 0 0 r F P 0 g s .53436 1.4749 m .5365 1.47421 L .60013 1.42016 L p .108 0 0 r F P 0 g s .60604 1.41773 m .60013 1.42016 L .5365 1.47421 L p .002 0 0 r F P 0 g s .5365 1.47421 m .54273 1.47162 L .60604 1.41773 L p .002 0 0 r F P 0 g s .61538 1.41289 m .60604 1.41773 L .54273 1.47162 L p 0 0 0 r F P 0 g s .36181 .40864 m .38646 .41429 L .31085 .39712 L p .16 0 0 r F P 0 g s .31085 .39712 m .28312 .39077 L .36181 .40864 L p .16 0 0 r F P 0 g s .54273 1.47162 m .55269 1.46639 L .61538 1.41289 L p 0 0 0 r F P 0 g s .62762 1.40508 m .61538 1.41289 L .55269 1.46639 L p 0 0 0 r F P 0 g s .59299 .54909 m .61755 .57564 L .56904 .53574 L p .624 .105 0 r F P 0 g s .56904 .53574 m .53768 .50344 L .59299 .54909 L p .624 .105 0 r F P 0 g s .55269 1.46639 m .5659 1.45786 L .62762 1.40508 L p 0 0 0 r F P 0 g s .64217 1.39377 m .62762 1.40508 L .5659 1.45786 L p 0 0 0 r F P 0 g s .68174 .66332 m .69473 .68857 L .66082 .64227 L p .732 .291 0 r F P 0 g s .66082 .64227 m .64036 .60672 L .68174 .66332 L p .732 .291 0 r F P 0 g s .38646 .41429 m .41375 .42338 L .34191 .40718 L p .301 0 0 r F P 0 g s .34191 .40718 m .31085 .39712 L .38646 .41429 L p .301 0 0 r F P 0 g s .74786 .81692 m .75121 .81767 L .76863 .88814 L p .806 .516 0 r F P 0 g s .75121 .81767 m .74786 .81692 L .72824 .76391 L p .78 .443 0 r F P 0 g s .72824 .76391 m .72581 .74901 L .75121 .81767 L p .78 .443 0 r F P 0 g s .5659 1.45786 m .58182 1.44542 L .64217 1.39377 L p 0 0 0 r F P 0 g s .65836 1.37854 m .64217 1.39377 L .58182 1.44542 L p .523 .85 .563 r F P 0 g s .78064 .98476 m .77637 .95388 L .76863 .88814 L p .727 .523 0 r F P 0 g s .77635 1.09228 m .77715 1.05157 L .78064 .98476 L p .752 .633 .071 r F P 0 g s .77637 .95388 m .78064 .98476 L .77715 1.05157 L 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.49356 m .44915 .46445 L .50577 .47621 L p .722 .191 0 r F P 0 g s .75693 1.14759 m .7605 1.09823 L .77715 1.05157 L p .934 .827 .377 r F P 0 g s .77344 1.00846 m .77715 1.05157 L .7605 1.09823 L p .973 .818 .401 r F P 0 g s .65925 1.34853 m .67835 1.31162 L .72517 1.27325 L p .88 .967 .643 r F P 0 g s .73867 1.23552 m .72517 1.27325 L .67835 1.31162 L p .926 .957 .635 r F P 0 g s .7043 .71741 m .71001 .74968 L .69237 .72587 L p .92 .537 .134 r F P 0 g s .69237 .72587 m .67832 .68209 L .7043 .71741 L p .92 .537 .134 r F P 0 g s .53768 .50344 m .56904 .53574 L .52525 .52781 L p .786 .286 0 r F P 0 g s .52525 .52781 m .48726 .49356 L .53768 .50344 L p .786 .286 0 r F P 0 g s .66082 .64227 m .67832 .68209 L .65287 .66184 L p .89 .468 .091 r F P 0 g s .65287 .66184 m .62731 .6152 L .66082 .64227 L p .89 .468 .091 r F P 0 g s .67835 1.31162 m .69583 1.26978 L .73867 1.23552 L p .926 .957 .635 r F P 0 g s .74947 1.19351 m .73867 1.23552 L .69583 1.26978 L p .958 .938 .62 r F P 0 g s .76722 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m .13252 .43162 L .20427 .39098 L p 0 0 0 r F P 0 g s .2064 .3903 m .2129 .3888 L .14159 .42931 L p .076 0 0 r F P 0 g s .14159 .42931 m .13473 .43091 L .2064 .3903 L p .076 0 0 r F P 0 g s .2129 .3888 m .22389 .38729 L .15331 .42759 L p .195 0 0 r F P 0 g s .15331 .42759 m .14159 .42931 L .2129 .3888 L p .195 0 0 r F P 0 g s .72634 .78134 m .71981 .80121 L .71147 .78513 L p .958 .622 .25 r F P 0 g s .71147 .78513 m .71001 .74968 L .72634 .78134 L p .958 .622 .25 r F P 0 g s .73133 .83867 m .74893 .85474 L .75291 .88912 L p .98 .693 .321 r F P 0 g s .74893 .85474 m .73133 .83867 L .72666 .8197 L p .97 .664 .278 r F P 0 g s .22389 .38729 m .23935 .38659 L .16996 .42656 L p .303 0 0 r F P 0 g s .16996 .42656 m .15331 .42759 L .22389 .38729 L p .303 0 0 r F P 0 g s .67832 .68209 m .69237 .72587 L .67524 .7125 L p .93 .548 .204 r F P 0 g s .67524 .7125 m .65287 .66184 L .67832 .68209 L p .93 .548 .204 r F P 0 g s .23935 .38659 m .25918 .3875 L .19152 .42699 L p .399 0 0 r F P 0 g s .19152 .42699 m .16996 .42656 L .23935 .38659 L p .399 0 0 r F P 0 g s .71096 1.22335 m .7231 1.17278 L .75693 1.14759 L p .979 .913 .601 r F P 0 g s .7605 1.09823 m .75693 1.14759 L .7231 1.17278 L p .992 .884 .578 r F P 0 g s .59914 .57306 m .62731 .6152 L .59771 .61105 L p .877 .444 .151 r F P 0 g s .59771 .61105 m .56233 .56708 L .59914 .57306 L p .877 .444 .151 r F P 0 g s .75973 1.04602 m .75427 .99165 L .76487 .96346 L p .991 .797 .416 r F P 0 g s .75122 .91717 m .76487 .96346 L .75427 .99165 L p .996 .768 .422 r F P 0 g s .76487 .96346 m .75122 .91717 L .75291 .88912 L p .98 .731 .325 r F P 0 g s .71001 .74968 m .71147 .78513 L .70252 .77316 L p .956 .613 .267 r F P 0 g s .70252 .77316 m .69237 .72587 L .71001 .74968 L p .956 .613 .267 r F P 0 g s .25918 .3875 m .28312 .39077 L .21779 .42962 L p .483 0 0 r F P 0 g s .21779 .42962 m .19152 .42699 L .25918 .3875 L p .483 0 0 r F P 0 g s .75291 .88912 m .73331 .85613 L .73133 .83867 L p .98 .693 .321 r F P 0 g s .73331 .85613 m .75291 .88912 L .75122 .91717 L p .988 .718 .372 r F P 0 g s .28312 .39077 m .31085 .39712 L .24845 .43514 L p .555 .039 0 r F P 0 g s .24845 .43514 m .21779 .42962 L .28312 .39077 L p .555 .039 0 r F P 0 g s .5365 1.47421 m .53436 1.4749 L .44827 1.47075 L p .471 .886 .928 r F P 0 g s .7231 1.17278 m .73165 1.1187 L .7605 1.09823 L p .992 .884 .578 r F P 0 g s .75973 1.04602 m .7605 1.09823 L .73165 1.1187 L p .998 .851 .553 r F P 0 g s .44827 1.47075 m .4505 1.47003 L .5365 1.47421 L p .471 .886 .928 r F P 0 g s .54273 1.47162 m .5365 1.47421 L .4505 1.47003 L p .558 .924 .94 r F P 0 g s .4505 1.47003 m .45709 1.46734 L .54273 1.47162 L p .558 .924 .94 r F P 0 g s .55269 1.46639 m .54273 1.47162 L .45709 1.46734 L p .638 .951 .943 r F P 0 g s .62731 .6152 m .65287 .66184 L .63064 .6593 L p .908 .508 .221 r F P 0 g s .63064 .6593 m .59771 .61105 L .62731 .6152 L p .908 .508 .221 r F P 0 g s .31085 .39712 m .34191 .40718 L .28305 .44416 L p .618 .116 .036 r F P 0 g s .28305 .44416 m .24845 .43514 L .31085 .39712 L p .618 .116 .036 r F P 0 g s .45709 1.46734 m .46778 1.4619 L .55269 1.46639 L p .638 .951 .943 r F P 0 g s .5659 1.45786 m .55269 1.46639 L .46778 1.4619 L p .707 .968 .935 r F P 0 g s .69237 .72587 m .70252 .77316 L .6939 .76659 L p .955 .612 .3 r F P 0 g s .6939 .76659 m .67524 .7125 L .69237 .72587 L p .955 .612 .3 r F P 0 g s .46778 1.4619 m .48219 1.45304 L .5659 1.45786 L p .707 .968 .935 r F P 0 g s .58182 1.44542 m .5659 1.45786 L .48219 1.45304 L p .763 .975 .921 r F P 0 g s .34191 .40718 m .37574 .42149 L .32101 .4572 L p .672 .185 .087 r F P 0 g s .32101 .4572 m .28305 .44416 L .34191 .40718 L p .672 .185 .087 r F P 0 g s .75427 .99165 m .74393 .93586 L .75122 .91717 L p .996 .768 .422 r F P 0 g s .73239 .87025 m .75122 .91717 L .74393 .93586 L p .991 .736 .422 r F P 0 g s .75122 .91717 m .73239 .87025 L .73331 .85613 L p .988 .718 .372 r F P 0 g s .73165 1.1187 m .73615 1.06181 L .75973 1.04602 L p .998 .851 .553 r F P 0 g s .75427 .99165 m .75973 1.04602 L .73615 1.06181 L p .999 .817 .525 r F P 0 g s .48219 1.45304 m .49986 1.44014 L .58182 1.44542 L p .763 .975 .921 r F P 0 g s .59982 1.42858 m .58182 1.44542 L .49986 1.44014 L p .809 .974 .901 r F P 0 g s .70843 .82339 m .72666 .8197 L .73133 .83867 L closepath p .978 .682 .378 r F P 0 g s .70843 .82339 m .71981 .80121 L .72666 .8197 L closepath p .975 .675 .371 r F P 0 g s .70843 .82339 m .71147 .78513 L .71981 .80121 L closepath p .973 .669 .369 r F P 0 g s .65287 .66184 m .67524 .7125 L .66045 .71125 L p .933 .565 .284 r F P 0 g s .66045 .71125 m .63064 .6593 L .65287 .66184 L p .933 .565 .284 r F P 0 g s .37574 .42149 m .41173 .44048 L .36164 .47463 L p .719 .249 .132 r F P 0 g s .36164 .47463 m .32101 .4572 L .37574 .42149 L p .719 .249 .132 r F P 0 g s .70843 .82339 m .73133 .83867 L .73331 .85613 L closepath p .979 .69 .39 r F P 0 g s .49986 1.44014 m .52023 1.42271 L .59982 1.42858 L p .809 .974 .901 r F P 0 g s .61922 1.40696 m .59982 1.42858 L .52023 1.42271 L p .846 .969 .878 r F P 0 g s .70843 .82339 m .70252 .77316 L .71147 .78513 L closepath p .971 .665 .372 r F P 0 g s .41173 .44048 m .44915 .46445 L .40414 .49671 L p .76 .307 .174 r F P 0 g s .40414 .49671 m .36164 .47463 L .41173 .44048 L p .76 .307 .174 r F P 0 g s .70843 .82339 m .73331 .85613 L .73239 .87025 L closepath p .98 .697 .404 r F P 0 g s .73615 1.06181 m .73623 1.00294 L .75427 .99165 L p .999 .817 .525 r F P 0 g s .74393 .93586 m .75427 .99165 L .73623 1.00294 L p .996 .781 .496 r F P 0 g s .52023 1.42271 m .54264 1.4004 L .61922 1.40696 L p .846 .969 .878 r F P 0 g s .63929 1.38032 m .61922 1.40696 L .54264 1.4004 L p .877 .959 .852 r F P 0 g s .67524 .7125 m .6939 .76659 L .68654 .76621 L p .952 .617 .341 r F P 0 g s .68654 .76621 m .66045 .71125 L .67524 .7125 L p .952 .617 .341 r F P 0 g s .44915 .46445 m .48726 .49356 L .44767 .52355 L p .797 .361 .211 r F P 0 g s .44767 .52355 m .40414 .49671 L .44915 .46445 L p .797 .361 .211 r F P 0 g s .54264 1.4004 m .56637 1.37298 L .63929 1.38032 L p .877 .959 .852 r F P 0 g s .65925 1.34853 m .63929 1.38032 L .56637 1.37298 L p .902 .947 .824 r F P 0 g s .70843 .82339 m .6939 .76659 L .70252 .77316 L closepath p .969 .663 .38 r F P 0 g s .74393 .93586 m .72862 .87949 L .73239 .87025 L p .991 .736 .422 r F P 0 g s .70843 .82339 m .73239 .87025 L .72862 .87949 L closepath p .98 .702 .419 r F P 0 g s .73623 1.00294 m .73167 .94297 L .74393 .93586 L p .996 .781 .496 r F P 0 g s .72862 .87949 m .74393 .93586 L .73167 .94297 L p .989 .743 .466 r F P 0 g s .48726 .49356 m .52525 .52781 L .49133 .55511 L p .829 .411 .246 r F P 0 g s .49133 .55511 m .44767 .52355 L .48726 .49356 L p .829 .411 .246 r F P 0 g s .56637 1.37298 m .59066 1.34039 L .65925 1.34853 L p .902 .947 .824 r F P 0 g s .67835 1.31162 m .65925 1.34853 L .59066 1.34039 L p .923 .932 .795 r F P 0 g s .52525 .52781 m .56233 .56708 L .53422 .5912 L p .858 .458 .277 r F P 0 g s .53422 .5912 m .49133 .55511 L .52525 .52781 L p .858 .458 .277 r F P 0 g s .70843 .82339 m .68654 .76621 L .6939 .76659 L closepath p .967 .663 .392 r F P 0 g s .59066 1.34039 m .6147 1.30273 L .67835 1.31162 L p .923 .932 .795 r F P 0 g s .69583 1.26978 m .67835 1.31162 L .6147 1.30273 L p .939 .915 .765 r F P 0 g s .73167 .94297 m .72239 .88281 L .72862 .87949 L p .989 .743 .466 r F P 0 g s .70843 .82339 m .72862 .87949 L .72239 .88281 L closepath p .979 .704 .433 r F P 0 g s .56233 .56708 m .59771 .61105 L .57544 .63147 L p .885 .502 .307 r F P 0 g s .57544 .63147 m .53422 .5912 L .56233 .56708 L p .885 .502 .307 r F P 0 g s .6147 1.30273 m .63769 1.26027 L .69583 1.26978 L p .939 .915 .765 r F P 0 g s .71096 1.22335 m .69583 1.26978 L .63769 1.26027 L p .953 .896 .733 r F P 0 g s .59771 .61105 m .63064 .6593 L .61417 .67546 L p .908 .545 .334 r F P 0 g s .61417 .67546 m .57544 .63147 L .59771 .61105 L p .908 .545 .334 r F P 0 g s .63769 1.26027 m .65888 1.21343 L .71096 1.22335 L p .953 .896 .733 r F P 0 g s .7231 1.17278 m .71096 1.22335 L .65888 1.21343 L p .964 .876 .7 r F P 0 g s .63064 .6593 m .66045 .71125 L .64965 .72257 L p .93 .586 .359 r F P 0 g s .64965 .72257 m .61417 .67546 L .63064 .6593 L p .93 .586 .359 r F P 0 g s .70843 .82339 m .68124 .77213 L .68654 .76621 L closepath p .966 .665 .405 r F P 0 g s .70843 .82339 m .67863 .78378 L .68124 .77213 L closepath p .965 .669 .419 r F P 0 g s .70843 .82339 m .67902 .79994 L .67863 .78378 L closepath p .964 .674 .432 r F P 0 g s .70843 .82339 m .6824 .81883 L .67902 .79994 L closepath p .965 .68 .443 r F P 0 g s .70843 .82339 m .68839 .83834 L .6824 .81883 L closepath p .966 .687 .451 r F P 0 g s .70843 .82339 m .69633 .85626 L .68839 .83834 L closepath p .968 .693 .456 r F P 0 g s .70843 .82339 m .70531 .87059 L .69633 .85626 L closepath p .971 .698 .456 r F P 0 g s .70843 .82339 m .71433 .87975 L .70531 .87059 L closepath p .974 .702 .452 r F P 0 g s .70843 .82339 m .72239 .88281 L .71433 .87975 L closepath p .977 .705 .444 r F P 0 g s .66045 .71125 m .68654 .76621 L .68124 .77213 L p .949 .626 .383 r F P 0 g s .68124 .77213 m .64965 .72257 L .66045 .71125 L p .949 .626 .383 r F P 0 g s .65888 1.21343 m .67754 1.16278 L .7231 1.17278 L p .964 .876 .7 r F P 0 g s .73165 1.1187 m .7231 1.17278 L .67754 1.16278 L p .972 .854 .665 r F P 0 g s .71566 .9374 m .71433 .87975 L .72239 .88281 L p .982 .741 .497 r F P 0 g s .72239 .88281 m .73167 .94297 L .71566 .9374 L p .982 .741 .497 r F P 0 g s .67754 1.16278 m .69302 1.10902 L .73165 1.1187 L p .972 .854 .665 r F P 0 g s .73615 1.06181 m .73165 1.1187 L .69302 1.10902 L p .978 .83 .627 r F P 0 g s .71244 .99544 m .71566 .9374 L .73167 .94297 L p .983 .774 .544 r F P 0 g s .73167 .94297 m .73623 1.00294 L .71244 .99544 L p .983 .774 .544 r F P 0 g s .69302 1.10902 m .70479 1.05295 L .73615 1.06181 L p .978 .83 .627 r F P 0 g s .73623 1.00294 m .73615 1.06181 L .70479 1.05295 L p .982 .803 .587 r F P 0 g s .70479 1.05295 m .71244 .99544 L .73623 1.00294 L p .982 .803 .587 r F P 0 g s .64965 .72257 m .68124 .77213 L .67863 .78378 L p .945 .638 .423 r F P 0 g s .6976 .91946 m .70531 .87059 L .71433 .87975 L p .972 .732 .516 r F P 0 g s .71433 .87975 m .71566 .9374 L .6976 .91946 L p .972 .732 .516 r F P 0 g s .67863 .78378 m .64417 .74552 L .64965 .72257 L p .945 .638 .423 r F P 0 g s .64417 .74552 m .67863 .78378 L .67902 .79994 L p .943 .652 .458 r F P 0 g s .61417 .67546 m .64965 .72257 L .64417 .74552 L p .924 .607 .424 r F P 0 g s .6795 .89093 m .69633 .85626 L .70531 .87059 L p .962 .718 .522 r F P 0 g s .70531 .87059 m .6976 .91946 L .6795 .89093 L p .962 .718 .522 r F P 0 g s .68538 .96927 m .6976 .91946 L .71566 .9374 L p .966 .755 .568 r F P 0 g s .71566 .9374 m .71244 .99544 L .68538 .96927 L p .966 .755 .568 r F P 0 g s .67902 .79994 m .64475 .77776 L .64417 .74552 L p .943 .652 .458 r F P 0 g s .64475 .77776 m .67902 .79994 L .6824 .81883 L p .944 .668 .486 r F P 0 g s .66348 .85496 m .68839 .83834 L .69633 .85626 L p .953 .702 .519 r F P 0 g s .69633 .85626 m .6795 .89093 L .66348 .85496 L p .953 .702 .519 r F P 0 g s .57544 .63147 m .61417 .67546 L .60555 .70921 L p .902 .577 .422 r F P 0 g s .64417 .74552 m .60555 .70921 L .61417 .67546 L p .924 .607 .424 r F P 0 g s .6824 .81883 m .65142 .8157 L .64475 .77776 L p .944 .668 .486 r F P 0 g s .65142 .8157 m .6824 .81883 L .68839 .83834 L p .947 .685 .506 r F P 0 g s .68839 .83834 m .66348 .85496 L .65142 .8157 L p .947 .685 .506 r F P 0 g s .13252 .43162 m .13473 .43091 L .08722 .53343 L p .387 .021 .215 r F P 0 g s .08722 .53343 m .08492 .53417 L .13252 .43162 L p .387 .021 .215 r F P 0 g s .13473 .43091 m .14159 .42931 L .0944 .53165 L p .443 .074 .242 r F P 0 g s .0944 .53165 m .08722 .53343 L .13473 .43091 L p .443 .074 .242 r F P 0 g s .14159 .42931 m .15331 .42759 L .10675 .52949 L p .497 .127 .268 r F P 0 g s .10675 .52949 m .0944 .53165 L .14159 .42931 L p .497 .127 .268 r F P 0 g s .6689 1.0192 m .68538 .96927 L .71244 .99544 L p .957 .773 .613 r F P 0 g s .71244 .99544 m .70479 1.05295 L .6689 1.0192 L p .957 .773 .613 r F P 0 g s .15331 .42759 m .16996 .42656 L .12442 .52763 L p .546 .176 .291 r F P 0 g s .12442 .52763 m .10675 .52949 L .15331 .42759 L p .546 .176 .291 r F P 0 g s .53422 .5912 m .57544 .63147 L .56345 .67538 L p .879 .548 .419 r F P 0 g s .60555 .70921 m .56345 .67538 L .57544 .63147 L p .902 .577 .422 r F P 0 g s .16996 .42656 m .19152 .42699 L .14741 .52672 L p .589 .22 .311 r F P 0 g s .14741 .52672 m .12442 .52763 L .16996 .42656 L p .589 .22 .311 r F P 0 g s .60555 .70921 m .64417 .74552 L .64475 .77776 L p .92 .63 .476 r F P 0 g s .19152 .42699 m .21779 .42962 L .17556 .52741 L p .628 .261 .328 r F P 0 g s .17556 .52741 m .14741 .52672 L .19152 .42699 L p .628 .261 .328 r F P 0 g s .65814 .92689 m .6795 .89093 L .6976 .91946 L p .949 .731 .574 r F P 0 g s .6976 .91946 m .68538 .96927 L .65814 .92689 L p .949 .731 .574 r F P 0 g s .49133 .55511 m .53422 .5912 L .51864 .64454 L p .856 .518 .414 r F P 0 g s .56345 .67538 m .51864 .64454 L .53422 .5912 L p .879 .548 .419 r F P 0 g s .64854 1.06844 m .6689 1.0192 L .70479 1.05295 L p .947 .788 .653 r F P 0 g s .70479 1.05295 m .69302 1.10902 L .64854 1.06844 L p .947 .788 .653 r F P 0 g s .21779 .42962 m .24845 .43514 L .20857 .53032 L p .663 .299 .344 r F P 0 g s .20857 .53032 m .17556 .52741 L .21779 .42962 L p .663 .299 .344 r F P 0 g s .44767 .52355 m .49133 .55511 L .47203 .61707 L p .831 .489 .407 r F P 0 g s .51864 .64454 m .47203 .61707 L .49133 .55511 L p .856 .518 .414 r F P 0 g s .24845 .43514 m .28305 .44416 L .24593 .53596 L p .696 .334 .358 r F P 0 g s .24593 .53596 m .20857 .53032 L .24845 .43514 L p .696 .334 .358 r F P 0 g s .28305 .44416 m .32101 .4572 L .28703 .54479 L p .726 .367 .371 r F P 0 g s .28703 .54479 m .24593 .53596 L .28305 .44416 L p .726 .367 .371 r F P 0 g s .40414 .49671 m .44767 .52355 L .42457 .59326 L p .807 .46 .4 r F P 0 g s .47203 .61707 m .42457 .59326 L .44767 .52355 L p .831 .489 .407 r F P 0 g s .4505 1.47003 m .44827 1.47075 L .34789 1.4014 L p .645 .821 .967 r F P 0 g s .34789 1.4014 m .35021 1.40066 L .4505 1.47003 L p .645 .821 .967 r F P 0 g s .45709 1.46734 m .4505 1.47003 L .35021 1.40066 L p .683 .83 .954 r F P 0 g s .32101 .4572 m .36164 .47463 L .3311 .55715 L p .754 .399 .382 r F P 0 g s .3311 .55715 m .28703 .54479 L .32101 .4572 L p .754 .399 .382 r F P 0 g s .36164 .47463 m .40414 .49671 L .37726 .57328 L p .781 .43 .391 r F P 0 g s .42457 .59326 m .37726 .57328 L .40414 .49671 L p .807 .46 .4 r F P 0 g s .37726 .57328 m .3311 .55715 L .36164 .47463 L p .781 .43 .391 r F P 0 g s .62481 1.11617 m .64854 1.06844 L .69302 1.10902 L p .935 .8 .687 r F P 0 g s .69302 1.10902 m .67754 1.16278 L .62481 1.11617 L p .935 .8 .687 r F P 0 g s .64475 .77776 m .60607 .7572 L .60555 .70921 L p .92 .63 .476 r F P 0 g s .60607 .7572 m .64475 .77776 L .65142 .8157 L p .92 .654 .517 r F P 0 g s .35021 1.40066 m .35716 1.39793 L .45709 1.46734 L p .683 .83 .954 r F P 0 g s .46778 1.4619 m .45709 1.46734 L .35716 1.39793 L p 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The text fails to mention that this integral equals ", Cell[BoxData[ \(TraditionalForm\`2 \[Pi]\ y\&_\ L\)]], " which is the the surface area of a cylinder whose radius is the average \ ", Cell[BoxData[ \(TraditionalForm\`y\)]], "-coordinate of the curve (", Cell[BoxData[ \(TraditionalForm\`y\&_\)]], ") and whose height is the length of the curve (", Cell[BoxData[ \(TraditionalForm\`L\)]], "). We computed ", Cell[BoxData[ \(TraditionalForm\`y\&_ = \[Pi] - 4\/3\)]], "in the previous problem 31, and ", Cell[BoxData[ \(TraditionalForm\`L = 4\)]], " in problem 17. Therefore, the area is ", Cell[BoxData[ \(TraditionalForm \`A\ = \ \(2 \( \[Pi](\[Pi] - 4\/3)\) \((4)\) = 8 \[Pi]\^2 - \(32\/3\) \[Pi]\)\)]], "." }], "Text"], Cell[BoxData[ \(x[t_] := \ Cos[t]; \ y[t_] := \ t + Sin[t]; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(A\ = \[Integral]\_0\%\[Pi] 2 \[Pi]\ y[t] \(\@\(\(x'\)[t]\^2 + \(y'\)[t]\^2\)\) \[DifferentialD]t\ \)\)], "Input"], Cell[BoxData[ \(\(-\(\(32\ \[Pi]\)\/3\)\) + 8\ \[Pi]\^2\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["54.", "Subsubsection"], Cell[TextData[{ "a) Plot the curve ", Cell[BoxData[ \(TraditionalForm\`x = e\^t + sin(2 t)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y = e\^t + cos(t\^2)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(-\[Pi]\) \[LessEqual] t \[LessEqual] \[Pi]/4\)]], "." }], "Text"], Cell[BoxData[ \(x[t_] := \ E\^t + Sin[2 t]; \ y[t_] := \ E\^t + Cos[t\^2]; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(p = ParametricPlot[{x[t], y[t]}, {t, \(-\[Pi]\), \[Pi]/4}, AspectRatio 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\(yp'\)[t]/\(x'\)[t] // Simplify; \nypp[t]\)], "Input"], Cell[BoxData[ \(\(1\/\((E\^t + 2\ Cos[2\ t])\)\^3\(( \((E\^t + 2\ Cos[2\ t])\)\ \((E\^t - 4\ t\^2\ Cos[t\^2] - 2\ Sin[t\^2])\) + \((\(-E\^t\) + 4\ Sin[2\ t])\)\ \((E\^t - 2\ t\ Sin[t\^2])\))\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ypp[\(-\[Pi]\)/4] // Simplify\)], "Input"], Cell[BoxData[ \(\(-\(1\/4\)\)\ E\^\(\[Pi]/2\)\ \((16 + \[Pi]\^2\ Cos[\[Pi]\^2\/16] + 8\ Sin[\[Pi]\^2\/16] + 2\ \[Pi]\ Sin[\[Pi]\^2\/16] + 8\ E\^\(\[Pi]/4\)\ \[Pi]\ Sin[\[Pi]\^2\/16])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[%]\)], "Input"], Cell[BoxData[ \(\(-77.2082507712387222`\)\)], "Output"] }, Open ]], Cell[TextData[{ "c) Find the equation of the tangent line at ", Cell[BoxData[ \(TraditionalForm\`t\_0 = \(-\[Pi]\)/4\)]], ".\n\nThe slope was found in part b): ", Cell[BoxData[ \(TraditionalForm\`m = 2.99294\)]], ". The point on the curve corresponding to ", Cell[BoxData[ \(TraditionalForm\`t\_0 = \(-\[Pi]\)/4\)]], " is ", Cell[BoxData[ \(TraditionalForm \`\((a, b)\) = \((e\^\(\(-\[Pi]\)/4\) + sin(\(-\[Pi]\)/2), \)\)]], Cell[BoxData[ \(TraditionalForm\`\(e\^\(\(-\[Pi]\)/4\) + cos(\[Pi]\^2/16))\)\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(a = N[x[\(-\[Pi]\)/4]]\), \(b = N[y[\(-\[Pi]\)/4]]\)}], "Input"], Cell[BoxData[ \(\(-0.544061872234003729`\)\)], "Output"], Cell[BoxData[ \(1.27164264518800496`\)], "Output"] }, Open ]], Cell[TextData[{ "The equation of the tangent line is: ", Cell[BoxData[ \(TraditionalForm\`y - b = m(x - a)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(y == \ b\ + \ m \((x - a)\) // Simplify\)], "Input"], Cell[BoxData[ \(y == \(2.89998637022126981`\[InvisibleSpace]\) + 2.99293850228289173`\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(tl = \ Plot[b\ + \ m \((x - a)\), {x, \(-2\), 2}, PlotStyle \[Rule] RGBColor[0, 0, 1]]\)], "Input"], 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