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# Class Note for MATH 250A with Professor Bergevin at UA

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Date Created: 02/06/15

MATH 250a Fall Semester 2007 Section 2 J M Cushing Thursday October 18 httprnathariz0naeducushing250ahtrnl Lengths 0f Curves What s the length of the curve Lengths 0f Curves What s the length of the curve Approximate length by curve of broken line segments Lengths 0f Curves What s the length of the curve Approximate length by curve of broken line segments Lengths 0f Curves What s the length of the curve Approximate length by curve of broken line segments Lengths 0f Curves What s the length of the curve Approximate length by curve of broken line segments Oi xi17 yi1 xiayi yi fx Where fX x2 Lengths of Curves What s the length of the curve Approximate length by curve of broken line segments Oi xi17 yi1 ALxi1 xi2 yi1yi2 Pythagoras Theorem xiayi yi fx Where fX x2 Lengths of Curves What s the length of the curve Approximate length by curve of broken line segments Oi xi17 yi1 xiayi yi fx Where fX x2 AL xi1 xi2 fxi1 fxi2 Assume equally spaced xi so that xi1 xi constant 2 Ax Lengths of Curves What s the length of the curve Approximate length by curve of broken line segments Oi xi17 yi1 xiayi yi fx Where fX x2 AL Ax2 fx Ax fX2 Assume equally spaced xi so that xi1 xi constant 2 Ax Lengths 0f Curves What s the length of the curve Approximate length by curve of broken line segments Oi xi17 yi1 x AL 0 xiayi yi fx Where fX x2 fltxiAx fltxgtZ Ax Ax AL Ax2 fxl Ax fx2 1 Lengths 0f Curves What s the length of the curve Approximate length by curve of broken line segments Oi xi17 yi1 x AL 0 xiayi yi fx Where fX x2 fltxiAx fltxgtZ Ax Ax z f39X AL Ax2 fxl Ax fx2 1 Lengths 0f Curves What s the length of the curve Approximate length by curve of broken line segments Oi xi17 yi1 xiayi yi fx Where fX x2 Arm1f39xi2 Ax Lengths 0f Curves What s the length of the curve L z EAL z Z 1f39xi2 Ax where f x x2 Lengths 0f Curves What s the length of the curve L z EAL z Z 1f39xi2 Ax where f x x2 X Pass the number of line segments to infinity Lengths 0f Curves What s the length of the curve Ljol1f39x2 dx where f x x2 Lengths 0f Curves y x What s the length of the curve Ljol1f39x2 dx where f x x2 In general the length of a curve y f x between points a f 61 and I f 19 is L E 1f39x2 dx Lengths 0f Curves What s the length of the curve Lj1 14x2 dx 0 where f x x2 Lengths 0f Curves What s the length of the curve LI114x2 dx2j1 lx2 dx 0 0 4 1 Lengths 0f Curves What s the length of the curve LI114x2 0421 lx2 dx 0 0 4 Table VI 30 and 29 X 1 1 1 1 1 I x2dx x x2 ln x x2 V4 2 V4 8 V4 Lengths 0f Curves y x What s the length of the curve LI114x2 0421 lx2 dx 0 0 4 Table VI 30 and 29 1 x I lx2dxlx lx2 llnx lx2 V4 2 V4 8 V4 1 1 1 1 1 Lz x V x2 1n x x2 2 4 8 4 0 Lengths 0f Curves y x What s the length of the curve LI114x2 dx2j1 lx2 dx 0 0 4 Table VI 30 and 29 1 x I lx2dxlx lx2 llnx lx2 V4 2 V4 8 V4 1 1 1 1 1 Lz x V x2 1n x x2 2 4 8 4 1 1 1 J5 1n2 5 1n 1 8 4J 8 2 0 Lengths 0f Curves y x What s the length of the curve LI114x2 dx2r lx2 dx 0 0 4 Table VI 30 and 29 1 x I lx2dxlx lx2 llnx lx2 V4 2 V4 8 V4 1 1 1 1 1 L x x2 1n x x2 2 4 8 4 1 J5 1 1 1n2 5 ln 1 407395 8 4 8 2 J 0 Lengths amp Areas in Polar Coordinates Sorne curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and y Lengths amp Areas in Polar Coordinates Sorne curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Location of a point not at the origin is determined x y by its rectangular coordinates Lengths amp Areas in Polar Coordinates Sorne curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Location of a point not at the origin is determined by its rectangular coordinates or equally as well by its polar coordinates Lengths amp Areas in Polar Coordinates Sorne curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Relationship is from basic trigonometry x rcos6 y rsin6 r 7rlt6 7r Lengths amp Areas in Polar Coordinates Some curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Relationship is from basic trigonometry xrcos6 yrsin6 I lt6 y 6 These permit a change to polar coordinates x in mathematical expressions expressed in rectangular coordinates Lengths amp Areas in Polar Coordinates Sorne curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Relationship is from basic trigonometry xrcos6 yrsin6 7 lt y lt6 6 Examples x A circle in rectangular coordinates x2 y2 4 Lengths amp Areas in Polar Coordinates Sorne curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Relationship is from basic trigonometry xrcos6 yrsin6 7 lt y lt6 6 Examples x A circle in rectangular coordinates x2 y2 4 in polar coordinates is r cos 6 2 r sin 602 4 Lengths amp Areas in Polar Coordinates Some curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Relationship is from basic trigonometry xrcos6 yrsin6 7 lt y lt6 6 Examples x A circle in rectangular coordinates x2 y2 4 2 2 in polar coordinates is r cos 6 r sm 6 4 or simply r 2 Lengths amp Areas in Polar Coordinates Some curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Relationship is from basic trigonometry xrcos6 yrsin6 r y 7r lt 6 S 7r 6 Examples x A circle in rectangular coordinatesl x2 y2 4 2 in polar coordinates is r cos 6 r sm 6 4 or simply Polar coordinates give simpler description Lengths amp Areas in Polar Coordinates Sorne curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Relationship is from basic trigonometry xrcos6 yrsin6 7 lt y lt6 6 Examples x The parabola in rectangular coordinates y x2 Lengths amp Areas in Polar Coordinates Sorne curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Relationship is from basic trigonometry xrcos6 yrsin6 7 lt y lt6 6 Examples x The parabola in rectangular coordinates y x2 2 in polar coordinates is r Sin 6 r cos 6 Lengths amp Areas in Polar Coordinates Sorne curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Relationship is from basic trigonometry xrcos6 yrsin6 7 lt y lt6 6 Examples x The parabola in rectangular coordinates y x2 2 in polar coordinates is r Sin 6 r cos 6 sin 6 or r20 and r 2 cos 6 Lengths amp Areas in Polar Coordinates Some curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Relationship is from basic trigonometry xrcos6 yrsin6 7 lt y lt6 6 Examples x The parabola in rectangular coordinates y x2 2 in polar coordinates is r sm 6 r cos 6 orr0 and r sng cos 6 Rectangular coordinates give simpler description Lengths amp Areas in Polar Coordinates Sorne curves and areas are better described in polar coordinates r and 6 rather than rectangular coordinates x and 2 Relationship is from basic trigonometry xrcos6 yrsin6 I lt6 y 6 Convert rectangular to polar coordinates x r le y2 6 arctanx X Convert polar to rectangular coordinates Lengths amp Areas in Polar Coordinates sin 6 r24 and r 2 cos 6 are examples of curves described in polar coordinates by equations of the form rf9 Lengths amp Areas in Polar Coordinates sin 6 r24 and r 2 cos 6 are examples of curves described in polar coordinates by equations of the form rf9 The rectangular coordinates are then xf6cos6 yf6sin6 Lengths amp Areas in Polar Coordinates sin 6 r24 and r 2 cos 6 are examples of curves described in polar coordinates by equations of the form rf9 The rectangular coordinates are then xf6cos6 yf6sin6 This is an example of a parameteric representation of a curve 6 969 y y9 Lengths amp Areas in Polar Coordinates Example rzka OSQltw radial distance is proportional to polar angle Lengths amp Areas in Polar Coordinates Example rzka OSQltw length Lengths amp Areas in Polar Coordinates Example rzka OSQltw length angular rneasure e g radians Lengths amp Areas in Polar Coordinates Example rzga OSQltw length angular rneasure e g radians constant of proportionality lengthradian Lengths amp Areas in Polar Coordinates Example r 6 O S 6 lt oo Lengths amp Areas in Polar Coordinates Example r 6 O S 6 lt oo invisible coefficient 1 has units that account for a balance of unites on each side of equation Lengths amp Areas in Polar Coordinates Example y r 6 O S 6 lt oo a spiral Lengths amp Areas in Polar Coordinates Example y r 6 O S 6 lt oo Parameteric representation x 600s6 y 6sin6 0 S 6 lt oo a spiral Lengths amp Areas in Polar Coordinates Example y r 6 O S 6 lt oo Parameteric representation x cos y 6sin6 0 S 6 lt oo Description in rectangular coordinates y x a spiral While not impossible is much more complicated Lengths amp Areas in Polar Coordinates Problem What is the length of the section I 6 0 S 6 S 27 of the spiral y 5 4 Parameteric representation x 600s6 y 6sin6 O S 6 S 27 Lengths amp Areas in Polar Coordinates A digression Formula for length of curve in rectangular coordinates y fx is Lj1f39x2 dx Lengths amp Areas in Polar Coordinates A digression Formula for length of curve in rectangular coordinates y fx is Lj1f39x2 dx What is the formula when given a parametric representation of the curve x 969 y y9 aSQS Lengths amp Areas in Polar Coordinates A digression Formula for length of curve in rectangular coordinates y fx is Lj1f39x2 dx What is the formula when given a parametric r esentation of the curve x 969 y y9 aSQS We get the formula by changing variables om x to 6 in the integral Lengths amp Areas in Polar Coordinates A digression Formula for length of curve in rectangular coordinates y fx is Lj1f39x2 dx What is the formula when given a p rametric representation of the curve e need to change the Integrand Lengths amp Areas in Polar Coordinates A digression Formula for length of curve in rectangular coordinates y fx is Lj1f39x2 dx What is the formula when given a parametr39 representation of the curve XX9 y yt9 0536 We get the formula by changing riables om x to 6 in the integral We 11 d to change the 0 Differential Lengths amp Areas in Polar Coordinates A digression Formula for length of curve in rectangular coordinates y fx is Lj1f39x2 dx What is the formula when giV a parametric representation of the curve x 969 yy9 aSQS We get the formula by ch ging variables om x to 6 in the integral e need to change the 0 lntegr nd 0 Diffe ential Limits of integration Lengths amp Areas in Polar Coordinates A digression L 1f39x2 dx By the Chain Rule d y d6 dx d6 Lengths amp Areas in Polar Coordinates A digression iL 1fxx 2aa mawmmm c d6 dxd6 dydyd6 mmme Lengths amp Areas in Polar Coordinates A digression L 1f39x2 dx By the Chain Rule d y d6 dx d6 dydt9 309 dx dxd x396 Lengths amp Areas in Polar Coordinates A digression L 1f39x2 dx 2 L 2 I 1 y 6 359 By the Chain Rule d1 d6 dx d6 dydt9 309 dx dxdt9 x396 Lengths amp Areas in Polar Coordinates A digression L 1f39x2 dx L 1 1 W 2 x396 3 x396 gt dx x396d6 Lengths amp Areas in Polar Coordinates A digression L 1f39x2 dx y39lt6gt2 L I 1x6x6d6 3 x396 gt dx x396d6 Lengths amp Areas in Polar Coordinates A digression L 1f39x2 dx szf 983 x 6d6 6 limits Lengths amp Areas in Polar Coordinates A digression NOT x limits a common student error Lengths amp Areas in Polar Coordinates A digression L 1f39x2 dx szf 983 x 6d6 Lengths amp Areas in Polar Coordinates A digression L 1f39x2 dx L j 18 x3962 d6 Lengths amp Areas in Polar Coordinates A digression L 1f39x2 dx L 3 3692 y39a2 d6 0 Lengths amp Areas in Polar Coordinates Problem What is the length of the section I 6 0 S 6 S 27 of the spiral y Parameteric representation 5 x2600s6 y6sin6 036327 L I x396 2 y3962 d6 4 Lengths amp Areas in Polar Coordinates Problem What is the length of the section I 6 0 S 6 S 27 of the spiral y Parameteric representation 5 x2600s6 y6sin6 4 x396 2 y3962 d6 Lengths amp Areas in Polar Coordinates Problem What is the length of the section I 6 0 S 6 S 27 of the spiral Parameteric representation x2600s6 y6sin6 x396 2 y3962 d6 L 6 sin 6 cos 62 6 cos 6 sin 602 d6 Lengths amp Areas in Polar Coordinates Problem What is the length of the section I 6 0 S 6 S 27 of the spiral Parameteric representation x 600s6 y 6sin6 O S 6 S 27 x L 3 x396 2 y3962 d6 06 L L 6 sin6 cos 62 6c056 sin 602 d6 szomxHQZ d6 Lengths amp Areas in Polar Coordinates Problem What is the length of the section I 6 0 S 6 S 27 of the spiral y Parameteric representation 5 x2600s6 y6sin6 036327 szoz 102d6 Fundamental Theorem or numerical approximation 4 Lengths amp Areas in Polar Coordinates Problem What is the length of the section I 6 0 S 6 S 27 of the spiral y Parameteric representation 5 x600s6 y6sin6 036327 szOZ KHaz d6 From Table VI 30 and 28 L6162 1ni9192 4 27 0 Lengths amp Areas in Polar Coordinates Problem What is the length of the section I 6 0 S 6 S 27 of the spiral y Parameteric representation 5 x2600s6 y6sin6 036327 szOZ KHaz d6 From Table VI 30 and 28 L6162 1ni9192 L 7nl47r2 1n27rl47r2 4 27 0 Lengths amp Areas in Polar Coordinates Problem What is the length of the section I 6 0 S 6 S 27 of the spiral y Parameteric representation 5 x2600s6 y6sin6 036327 szOZ KHaz d6 From Table VI 30 and 28 L6162 1ni9192 L 7nl47r2 1n27rl47r2z 2126 4 27 0 Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y Break into approximating circular sectors Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y Break into approximating circular sectors make circular Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y Break into approximating circular sectors Area of a circular sector Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y Break into approximating circular sectors Area of a circular sector It is a fraction of the area TEF2 of the circle Which fraction Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y Break into approximating circular sectors Area of a circular sector It is a fraction of the area TEF2 of the circle Which fraction Same fraction that A6 is of 27 A6 27 Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y Break into approximating circular sectors Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y Break into approximating circular sectors Area of circular sector an A g rz zlrzAQ 27 Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y Break into approximating circular sectors 5 4 1 AA z Area of Circular sector 2 r2A6l 2 Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y 6quot Break into approximating circular sectors 1 2 AA z Area of Circular sector 2 Er A6 AA z azm Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y Break into approximating circular sectors 5 4 1 AA z Area of Circular sector 2 r2A6l 2 1 2 AAzEQ A6 27239 1 Total area A z ZEQZAQ 60 Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y Break into approximating circular sectors 5 4 1 AA z Area of circular sector 2 EFZAQ AA z azm 27239 1 Total area A z ZEQZAQ 60 27r 1 2 Pass to the limit A 2 I0 36 d6 Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y 2 Azj Qazda 0 2 4 Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y 6 2 l A j 62d6 4 0 2 27r Lengths amp Areas in Polar Coordinates Problem Find the area enclosed by the spiral r 6 O S 6 S 27 and the xaXis y 6 2 l A j 62d6 4 0 2 27r Lengths amp Areas in Polar Coordinates In general the area enclosed by curve r f 9 a S 9 S is Areas of approximating circular sectors Lengths amp Areas in Polar Coordinates In general the area enclosed by curve r f 9 a S 9 S is B 1 A lim 2 r2A6 a 2 A jg 9W9

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