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# Class Note for MATH 1432 at UH 2

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Date Created: 02/06/15
Lecture 16 Section 98 Arc Length and Speed Jiwen He Department of Mathematics University of Houston jiwenheCQmath uh edu http math uh eduNj iwenheMath1432 What is the length of this curve F1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 Arc Length Arc Lenh Examles Arc Length Formulas Arc Length Formulas Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 ll Arc Length Arc Lenh Examles Arc Length Formulas Arc Length Formulas Let C xtyt t e I Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Arc Length Arc Lenh Examles Arc Length Formulas Arc Length Formulas Let C xtyt t e I The length of C is b LC X t2 y t2dt Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Arc Length Arc Lenh Examles Arc Length Formulas Arc Length Formulas Let C xtyt t e I The length of C is b LC X t2 y t2dt Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Arc Length Arc Lenh Examles Arc Length Formulas Arc Length Formulas Let C xtyt t e I The length of C is i b a polygonal path inscribed in the curve C Xt 2 i 2 dt Po dP1 Pi Vixen Xt12 yt yt12 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Arc Length Arc Lenh Examles Arc Length Formulas Arc Length Formulas Let C xtyt t e I The length of C is i b a polygonal path inscribed in the curve C Xt 2 i 2 dt Po dP1 Pi Vixen Xt12 yt yt12 mo xvi1 2 you yon1 2 M ti ti l ti ti l Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Arc Length Arc Length Formulas Arc Length Formulas Let C xtyt t e I The length of C is i b a polygonal path inscribed in the curve C Xt 2 i 2 dt Po dP1 Pi Vixen Xt12 yt yt12 Vfwwxw1q2immyuioruity ti ti l ti ti l Vw vw mi Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Arc Length Arc Length Formulas Arc Length Formulas Let C xtyt t e I The length of C is i b apolygonal path inscribed inthe curveC Xt 2 l 2 dt Po dP1 Pi We xt112 yt yt112 Vfwwxw1q2 mmyuioruity ti ti 1 ti ti 1 Vw vw mi Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Arc Length Arc Length Formulas Arc Length Formulas Let C xtyt t e I The length of C is i b apolygonal path inscribed inthe curveC Xt 2 l 2 dt Po dP1 Pi We xt112 yt yt112 Vfwwxw1q2 mmyuioruity ti ti 1 ti ti 1 Vw vw mi LW dP0 P1 dPi 17Pi dPn 17Pn Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Arc Length Arc Length Formulas Arc Length Formulas Let C xtyt t e I The length of C is i b apolygonal path inscribed inthe curveC Xt 2 l 2 dt Po dP1 Pi We xt112 yt yt112 mo Ann 2 you yon1 2 M ti ti 1 ti ti 1 Vw vw mi LW dP0 P1 dPi 17Pi dPn 17Pn n 2 Wm WNW 1 1 i1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Arc Length Arc Length Formulas Arc Length Formulas Let C xtyt t e I The length of C is i b apolygonal path inscribed inthe curveC Xt 2 l 2 dt Po dP1 Pi We xt112 yt yt112 mo Ann 2 you yon1 2 M ti ti 1 ti ti 1 Vw vw mi LW dP0 P1 dPi 17Pi dPn 17Pn Z xxtnfwanfmiem 35AM 1 1 i1 j Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Arc Length Arc Length Formulas Arc Length Formulas i Let C xtyt t e I The length of C is b 2 2 apolygonalpaminscribedmmemc LC xt yt dt 0 We define the element of length d5 ds xt2 yt2 dt Po Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Arc Length Arc Length Formulas Arc Length Formulas i Let C xtyt t e I The length of C is b 2 2 apolygonalpaminscribedmmemc LC xt yt dt 0 We define the element of length d5 ds xt2 yt2 dt 0 The total arc length is LC ds ab X t2 yt2 dt Po Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 2 11 Parametrization by the Motion What is the length of this curve Arc Length and Speed Along a Plane Curve Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 3 11 Parametrization by the Motion 0 Imaging an object moving along the curve C What is the length of this curve Arc Length and Speed Along a Plane Curve Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 3 11 Parametrization by the Motion 0 Imaging an object moving along the curve C 0 Let rt Xtyt the position of the object at time t What sme39 gthomime 0 The velocity of the object at time t is var rm x ty39t Arc Length and Speed Along a Plane Curve Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 Parametrization by the Motion 0 Imaging an object moving along the curve C 0 Let rt Xtyt the position of the object at time t What sme39 gthomime 0 The velocity of the object at time t is var rm x ty39t Arc Length and Speed Along a Plane Curve 0 The speed of the object at time t is Va llvtll Viva M Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 Parametrization by the Motion N 0 Imaging an object moving along the curve C 0 Let rt Xtyt the position of the object at time t What sme39 gthomime 0 The velocity of the object at time t is var rm wryw Arc Length and Speed Along a Plane Curve 0 The speed of the object at time t is 2 2 Va llvtll dim M o The distance traveled by the object from time zero to any later time t is st ds OtXU2 y u2du Otvu du Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 Parametrization by the Motion N 0 Imaging an object moving along the curve C 0 Let rt Xtyt the position of the object at time t What sme39 gthomime 0 The velocity of the object at time t is vt vt wryw Arc Length and Speed Along a Plane Curve 0 The speed of the object at time t is 2 2 vt llvtll dim M o The distance traveled by the object from time zero to any later time t is st ds OtXU2 y u2du Otvu du dt 0 We have ds vt Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 Arc Length Arc Lenth Examles Length of the Arc on the Graph of y fX Length ofy fX X E 3 b Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 4 11 Arc Length Arc Lenth Examles Length of the Arc on the Graph of y fX Length ofy fX X E 3 b tangent The length of the arc on the graph from a to X is 5X f V 1 f t2 dt gr Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 4 11 Arc Length Arc Lenth Examies Length of the Arc on the Graph of y fX Length ofy fX X E 3 b The length of the arc on the graph from a to X is 5X f V 1 f t2 dt gr Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 4 11 Arc Length Arc Lenth Examies Length of the Arc on the Graph of y fX Length ofy fX X E 3 b The length of the arc on the graph from a to X is 5X f V 1 f t2 dt gr Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 4 11 Arc Length Arc Lenth Examies Length of the Arc on the Graph of y fX Length ofy fX X E 3 b The length of the arc on the graph from a to X is 5X f V 1 f t2 dt gr Proof Set Xt t yt ft t e a b Since X t 1 y t f t then 5X x t2 ix132 dt f i 1 f t2 dt H Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 4 11 Arc Length Arc Lenth Examies Length of the Arc on the Graph of y fX Length ofy fX X E 3 b The length of the arc on the graph from a to X is 5X f V 1 f t2 dt x gt d5 1 l f x2dx Set Xt t yt ft t e a b Since X t 1 y t f t then 5X x t2 ix132 dt f i 1 f t2 dt H Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 4 11 The length of the are on the graph from a to X is o The length of the parabolic arc fx 2 X2 X E 01 is given Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 4 11 X Length ofy fX X E a b The length of the are on the graph from a to X is 5X f V 1 f t2 dt x gt d5 1 l f X2dX o The length of the parabolic arc fX 2 X2 X E 01 is given 011 f X2dX Jiwen He University of Houston Math 1432 a Section 26626 Lecture 16 March 6 2008 4 11 X Length ofy fX X E a b The length of the are on the graph from a to X is 5X f V 1 f t2 dt x gt d5 1 l f X2dX o The length of the parabolic arc fX 2 X2 X E 01 is given fox1 f x2dxOlmdx Jiwen He University of Houston Math 1432 a Section 26626 Lecture 16 March 6 2008 4 11 X Length ofy fX X E a b The length of the are on the graph from a to X is 5X f V 1 f t2 dt x gt d5 1 l f X2dX o The length of the parabolic arc fX 2 X2 X E 01 is given Almdxzfolmdx XWiiIMXl W H Jiwen He University of Houston Math 1432 a Section 26626 Lecture 16 March 6 2008 4 11 X Length ofy fX X E a b The length of the are on the graph from a to X is 5X f V 1 f t2 dt x gt d5 1 l f X2dX o The length of the parabolic arc fX 2 X2 X E 01 is given Almdxzfolmdx Xix2lnxix2 iln2 H Jiwen He University of Houston Math 1432 a Section 26626 Lecture 16 March 6 2008 4 11 Arc Length Arc Lenth Examies Length of the Arc on the Graph of r Length of r p9 9 E 0475 rza 620 spiral of Archimedes Proof Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 5 11 I Polar aXIs J rza 620 spiral of Archimedes Proof The length of the arc on the graph from ozto is 9 s6 V WV W dt Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 511 th Exam les Length of the 39Polaraxis The length of the arc on the graph from oz to 9 is 9 s6 V pti2 W dt rza 620 spiral of Archimedes Proof Set Xt 0t cos t yt 0t sin t t 6 045 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 5 11 th Exam les Length of r p9 9 E 0475 The length of the arc on the graph from 04 to 9 is 9 s6 V pti2 W dt rza 620 spiral of Archimedes Proof Set Xt 0t cos t yt 0t sin t t 6 045 Since x ti2 M0 pti2 Na Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 5 11 th Exam Ies p 39Polaraxis The length of the arc on the graph from azto Qis i9 56 V ipti2 W dt rza 620 spiral of Archimedes Proof 1 Set Xt 0t cos t yt 0t sin t t 6 045 Since Xt2 yt2 pt2 pt2 then 58 x t2 ix132 dt pt2 p t2 dt 11 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 5 11 th Exam 1625 M Length of r p9 9 E 0475 The length of the arc on the graph from 04 to 9 is 9 s6 V pti2 W dt dsie WW WW d6 spiral of Archimedes Proof 1 Set Xt 0t cos t yt 0t sin t t 6 045 Since Xt2 yt2 pt2 pt2 then 58 x t2 ix132 dt pt2 p t2 dt 11 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 5 11 Length of r p9 9 E 0175 The length of the arc on the graph from 04 to 9 is 9 s6 V ipti2 W dt dsie WW WW d6 spiral of Archimedes Spiral of Archimedes r i9 9 Z O o The length of the arc r i9 9 E 027r is given Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 5 11 Exam Ies frp Length of r p9 9 E 0175 The length of the arc on the graph from 04 to 9 is 9 s6 We W dt dsie WW WW d6 spiral of Archimedes Spiral of Archimedes r i9 9 Z O o The length of the arc r i9 9 E 027r is given f pm 2 W d6 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 5 11 Exam Ies frp Length of r p9 9 E 0175 The length of the arc on the graph from 04 to 9 is 9 s6 We W dt dsie WW WW d6 spiral of Archimedes Spiral of Archimedes r i9 9 Z O o The length of the arc r i9 9 E 027r is given V W WW d6 2 V1 62 d6 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 5 11 Exam Ies frp Length of r p9 9 E 0175 The length of the arc on the graph from 04 to 9 is 9 s6 We W dt dsie WW WW d6 spiral of Archimedes Spiral of Archimedes r i9 9 Z O o The length of the arc r i9 9 E 027r is given V W WW d6 2 V1 62 d6 0 1 1 27t O Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 5 11 Exam Ies frp Length of r p9 9 E 0175 The length of the arc on the graph from 04 to 9 is 9 s6 We W dt dsie WW WW d6 spiral of Archimedes Spiral of Archimedes r i9 9 Z O o The length of the arc r i9 9 E 027r is given V W WW d6 2 V1 62 d6 0 27t O 2E9 182n8l 82 I Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 5 11 Arc Length Example Circle of Radius a L 27ra r22 r 2 r4sin0 1 2 40036 Circle in Polar Coordinates Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 6 11 e r 2 r4sin0 1 2 40036 Circle in Polar Coordinates r a O S 0 S 27r 27139 L f M2 W d6 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 6 11 QMDGL r4sin0 1 2 40036 Circle in Polar Coordinates r a O S 0 S 27r L 2Wp02 p 62d0 2Wa20d0 0 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 6 11 QMDGL r4sin0 1 2 40036 Circle in Polar Coordinates r a O S 0 S 27r L 27r V p02 p 02d0 27F V a2 0d0 27ra 0 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 6 11 r22 r 2 r4sin6 1392 40050 Circle in Polar Coordinates r2asin0 0 0 7r Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 6 11 r22 r 2 r4sin6 1392 40030 Circle in Polar Coordinates r2asin0 0 0 7r L for W W d6 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 6 11 Arc Length Example Circle of Radius a L 27ra mm W95 Circle in Polar Coordinates r2asin0 0 0 7r L for i W W d6 f 2asin 02 2a cos 0l2d0 r4sin6 1392 40030 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 6 11 Arc Length Example Circle of Radius a L 27ra mm W95 Circle in Polar Coordinates r2asin0 0 0 7r r4sin6 1392 40030 L for W W d6 2 2asin02 2ac050l2d022a d027ra 0 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 16 March 6 2008 6 11 Arc Length Example Limagon 27r polar axis 1 51 2 3 Limagon r 1 c056 The length of the cardioid r 1 c056 9 E 0 27 is given Jiwen He University of Houston Math 1432 Sectian 26626 Lecture 16 March 6 20018 T 11 Arc Length Example Limagon i 2 polar axis 1 51 2 3 27r Limagon r 1 c056 The length of the cardioid r 1 c056 9 E 0 27 is given 0277 V W 2 we 2 d6 Jiwen He University of Houston Math 1432 Sectian 26626 Lecture 16 March 6 20018 T 11 Arc Length Example Limagon i 2 polar axis 1 51 2 3 27r Limagon r 1 c056 The length of the cardioid r 1 c056 9 E 0 27 is given f p92 p62d6 207Tsin62 1 cos62d9 Jiwen He University of Houston Math 1432 Sectian 266126 Lecture 16 March 6 2008 T 11 Arc Length Example Limagon i 2 polar axis 1 51 2 3 27r Limagon r 1 c056 The length of the cardioid r 1 c056 9 E 0 27 is given 027Tp92 p62d6 207Tsin62 1 cos62d9 2Wmd6 H O Jiwen He University of Houston Math 1432 Sectian 266126 Lecture 16 March 6 2008 T 11 Arc Length Example Limagon i 2 polar axis 1 51 2 3 27r Limagon r 1 c056 The length of the cardioid r 1 c056 9 E 0 27 is given 027Tp6 2 p 62d6207Tsin62 1 cos62d6 2Wmd62W2sin6d6 H O O Jiwen He University of Houston Math 1432 Sectian 266126 Lecture 16 March 6 2008 T 11 Arc Length Example Limagon i 2 polar axis 1 51 2 3 27r Limagon r 1 c056 The length of the cardioid r 1 c056 9 E 0 27 is given 027Tp6 2 p 62d6207Tsin62 1 cos62d6 27TMd 2W2sin36d68 cosl 8 H 0 0 2 2 0 Jiwen He University of Houston Math 1432 Sectian 266126 Lecture 16 March 6 2008 T 11 Example Logarithmic spiral r aebo Example Logarithmic spiral r aebo or r s a special kind of spiral curve which often appears in nature Example Logarithmic spiral r aebo w s a special kind of spiral curve which often appears in nature 0 The polar equation of the curve is y wig9 or a 4 Lilhl A U i u mu 1 Example Logarithmic spiral r aebo s a special kind of spiral curve which often appears in nature 0 The polar equation of the w curve is r a or 4 7 71 l I A U i U link 12 o The spiral has the property that the b between the tangent and radial line at the point r76 is and a Thezvvmazh M2 mmm as pray Tiewshzvgtmew m in huge thew mm wth a Thezvvmazh M2 mmm as pray Tiewshzvgtmew m in huge thew mm wth a m zvvmazh M2quot mng 2 km mm The m g m hwme hm mm m mmmngbw mum m a 329 at 2 a 3 ig mz z S g gums 2 952 3 a 1m 29 a swig 2 as 3 in s 323 Him 5 391 22 a a 25 a Fa 233 133212 2633 a Thezvvmazh M2 mmm as pray Tiewshzvgtmew m in huge thew mm wth a m zvvmazh M2quot mng 2 km mm 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at the corners of a square a They start to craw c ockwrse at a constant rate each movmg toward rts nerghbor a At any mstant they rnark the corners of a square As the bugs get c oser to the orrgma square s center the new square they dehne rotates and dnnrnrshes m srze new m m u Four Bug mg One Anoth 0 Four bugs are at the corners of a square a They start to craw c ockwrse at a constant rate each movmg toward rts nerghbor a At any mstant they rnark the corners of a square As the bugs get c oser to the orrgma square s center the new square they dehne rotates and dnnrnrshes m srze a Each bug starts at a corner ofthe orgma umt square that rs l away from the orrgrn r e center and moves mward a ong the sprra wrth the angre Al 39ms to u Four Bug mg One Anoth 0 Four bugs are at the corners of a square a They start to craw c ockwrse at a constant rate each movmg toward rts nerghbor a At any mstant they rnark the corners of a square As the bugs get c oser to the orrgma square s center the new square they dehne rotates and dnnrnrshes m srze a Each bug starts at a corner ofthe orgma umt square that rs l away from the orrgrn r e center and moves mward a ong the sprra wrth the ang e 3 The sprra rnotron rs descnbed by g in I01 39ms to u Four Bug mg One Anoth 0 Four bugs are at the corners of a square 6 They start to craw c ockwrse at a constant rate each movmg toward rts nerghbor a At any mstant they rnark the corners of a square As the bugs get c oser to the orrgma square s center the new square they dehne rotates and dnnrnrshes m srze a Each bug starts at a corner ofthe orgma umt square that rs l away from the orrgrn r e center and moves mward a ong the sprra wrth the ang e Al The sprra rnotron rs descrrbed by gg in 0 1N5 The porar equatron or the path rs r l e g 9 3 0 H 39ms to u Four Bug mg One Anoth 0 Four bugs are at the corners of a square a They start to craw c ockwrse at a constant rate each movmg toward rts nerghbor a At any mstant they rnark the corners of a square As the bugs get c oser to the orrgma square s center the new square they dehne rotates and dnnrnrshes m srze a Each bug starts at a corner ofthe orgma umt square that rs l away from the orrgrn r e center and moves mward a ong the sprra wrth the ang e Al The sprra rnotron rs descrrbed by gg in 0 1N5 The porar equatron or the path rs r 1 9 a 3 0 Thetotar drstance covered H on rts path rs LC 39ms to u r M Length 0 AK Length a my NW N M Mr m

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