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# Class Note for MATH 1432 at UH 2

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This 9 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 21 views.

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Date Created: 02/06/15

LGCture 16Secti0n 98 Arc Length and Speed J iwen He 1 Arc Length 11 Arc Length Arc Length Formulas Pn l a polygonal path inscribed in the curve C P0 Arc Length Formulas Let C t E I 05ex The length of C is 1 mm aclt0 glow dt 7 Pi 171339 tz39 ti 1W WM M7514 at won n 2 m yltt 1gtrltti W 75 ti l 75 ti l acW y lttgt2Ati Z d P07P1gt dltP 1P gt dltPn1Pngt 2 w lttgt2wthth Muse De nition 1 c We de ne the element of length ds ds x mm y t2 dt 1 o The total arc length is l Lltcgt ds acmi det Arc Length and Speed Along a Plane Curve What is the length of this curve Parametrization by the Motion 0 Imaging an object moving along the curve 0 0 Let rt xtyt the position of the object at time t o The velocity of the object at tiInet is vt rt m ty tl Arc Length and Speed Along a Plane Curve 0 The speed of the object at time t is vt x t2 y t2l o The distance traveled by the object from time zero to any later time t is t t 5t d5 x u2 y u2du vu dul 0 0 0 We have d5 vt dtl 12 Examples Length of the Are on the Graph of y x VA tangent RV Length of y m z 6 11 The length of the are on the graph from a to z is smVHVMWt i 13 1f m dz Proof Set Mt t yt ft t 6 11 lex Since z t 1 y t f t then SMA3ww wwww1x1vwwt Example 2 o The length of the parabolic are x H z E 0 1 is given 1 2 1 11 dm V14m2dm 0 0 1 xiiz2ilnz1iz2o xEiln2E Length of the Are on the Graph of r 30 I Polar aXIS rza 620 spiral of Archimedes Length of 7 p0 H E 04 The length of the are on the graph from a to H is 0 39 A l m2 mm2 dt cw pan pm d0 Proof Set W ptcost yt 0tsint t 6 cm lex Since asm2 WW PM2 pm then Spiral of Archimedes 7 0 H 2 0 o The length of the arc 7 H H E 0 27139 is given 27r 27r lpi92 p02d0 102d0 0 0 27r O x102ln0102 Example Circle of Radius a L 27m mm A q WW r2 r 2 r45in9 r 4c056 Circle in Polar Coordinates 7 a 0 g 6 g 27T r2asin6 0 9 7T L02Wp02 p 02d0027T a20d027ra L F WW WW d6 7T 2asin02 2acos62d6 20L7T d6 27m Example Limagon 1 1 2 3 polar axis F511 2 3 27r Limagon 7 1 COSQ The length of the cardioid 7 1 cos 6 6 E 0 27139 is given 027T lp62 p 02d6 207T sin62 1 cos92d0 7T 7T 1 1 7T 2 x21 cos6d62 2sin 0d98 cos 6 8 0 0 2 2 0 Lagamhmm spual v azw a Example m gamma ma Lagamhmm Spual v az A logamlfvmc xpwl zqwngalar swam Wm xpwlxs a spsmal kmd Dfsyual curve whmh am appeals m name The polax aquan amas curve Is 7 as a a H lama The spud has the ympety mac mas magi a beam mas tangent and xadxal me at Lheymnc 7915 maximal and a mcmbquot Lagamhmm Spual 111 Mohan v az39 g 2 o Spual Muhum o The approach of a hawk to its prey Their sharpest View is at an angle to their direction of ight o The approach of an insect to a light source They are used to having the light source at a constant angle to their ight path 0 Starting at a point P and moving inward along the spiral with the angle 45 Let a be the straightline distance from P to the origin The spiral motion is described by dT E 7127 T0 a With 12 7 cot The polar equation of the path is T ae be t9 2 0 Length of the Logarithmic Spiral T ae be7 6 2 0 o The length of the logarithmic spiral T 6 9 9 2 07 is given 0 2 2 0 2 2 LC 0 939 pm d6 0 e79 e79 d6 me 9d9 ie 9lw 0 o The spiral motion T ae be t9 2 0 circles the origin an unbounded number of times without reaching it yet7 the total distance covered on this path is nite LC Ooo d3 a coslt157 with 45 cot 1 bi Four Bugs Chasing One Another Four Bugs Chasing One Another 0 Four bugs are at the corners of a square 0 They start to crawl Clockwise at a constant rate7 each moving toward its neighbor 0 At any instant7 they mark the corners of a square As the bugs get Closer to the original square7s center7 the new square they de ne rotates and diminishes in size 0 Each bug starts at a corner of the orginal unit square that is l away from the origin iiei7 center and moves inward along the spiral With the angle 45 g The spiral motion is described by 77 7 0 The polar equation of the path is 7 1 e 97 t9 2 0 The total distance covered on its path is LC 1 Outline Contents 1 Arc Length 1 1 1 Arc Length i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 1 12 Examples i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 3

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