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# Chapter 9.1 - 9.4 MTH 162

UM

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This 8 page Class Notes was uploaded by Jack Magann on Sunday April 19, 2015. The Class Notes belongs to MTH 162 at University of Miami taught by Dr. Bibby in Spring2015. Since its upload, it has received 123 views. For similar materials see Calculus 2 in Mathematics (M) at University of Miami.

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Date Created: 04/19/15

Calculus 2 chapters 91 92 Parametric equations Equations and curves in the XY plane can be represented with 1 y f x 2 x f y 3 F x y O implicit 4 Parametric Equation x fa Where t is referred to as a parameter y 90 This represents a point on a curve as shown by the below table x t2 For y t3 t x y Point at xy 0 0 00 1 1 1 11 1 1 11 2 4 8 48 2 4 8 48 Ex Graph without using a table x t 1 y t2 Through substitution y x2 easy enough to graph x sin t 2 y Sing t gt 3 x due to the domain ofthe sin function only from 1 to 1 t 3 x 6 gt y x2 domain of et is O lt t y 621 4 Eliminate the parameter find the equation of the graph x3 2t 3 x y23t 2t3x Iquot 3x E 2 E y 223 x 22 2 2x2 96 4 x 4 2 cos9 C0509 T y 1 2sin 9 Smog 3 2 5 square both sides and add them 2 2 1 so x 42y124 This is a circle When the parameter can t be easily done one needs to use differentiation dy Given x fa Then d y y 90 d f dx Ex x t2 t 2 1 F1nd the equatlon of the 11ne of tangent to the curve t3 3t at t 2 a Plug in t to find starting point x 4 y 2 at t 2 Point of tangency 42 dy d3 2 b dx 2t1 Att Z dx 3 3 Slopeoftangent c Find the equation for the tangent y 23x 4 gt y3x 10 Second derivatives x ft dzy d dy dy Givenyzga 5 dx dx Ex d2 x 9 sin 9 1 F1nd y of dx2 y 1 cos9 x 9 sin 9 d y 5M9 so now 39 9 dx 1 cos0 39 L 1 cos0 2 d3quot I 2 2 2 d y E dl 1 c050c050 51n 0 cosQ cos 0 51n 0 cosQ l 1 dx2 d0 1 cost92 1 cost92 1 cost92 1 c050 2 dy 1 d 3 dQ 1 c056 1 1 1 dx2 1 c050 1 c050 1 c050 1 cos 192 Parametric equations and Arc length For calculating arc length we already know two formulas 2 1yfxanSb sf1 dx b d 2 2 xf aSySb sfa 1 dy There is a third using parametric equations x xt 3 313 aStSb 3fo 22dt Ex 1 Find the length of the arc for fl if 31522 914 212 4152 s f01V9t4 4t2dt join912 4dt u 912 4 du 18t s iflxEdu i3u 1 i9t2 41 i 13V13 18 18 0 18 3 0 27 0 27 x9 sin9 y 1 COS 9 is an example of a cycloid Area under a curve Remember b For regular equatlons fa f x dx IS the area under the curve Ex Find the area under one arch of the cycloid x i 9 5m 9 y 1 cos 9 Area f9902nydx y 1 c059 dx 1 cos 9 2 f0 1 cos 6X1 cos 9d9 2 f0 1 2 c059 cos 9d9 f02n1 ZCOSH 1 c0529 d9 foznG ZCOSH c0529 d9 3 1 2 3 9 2 sm9 sm29 X2739 2 37 square umts 2 4 0 2 Calculus 2 Chapters 93 94 Rectangular coordinates x y Polar coordinates 139 Consider Point P x y in rectangular or r 9 in polar Conversion From 1quot 9 to x y c059 2 gt x rcosH sin9 2 gt y rsinH x2 y2 2 From x y to r 9 r W cost9xz y2 sin9 2 tan9 For tangent it could be in multiple quadrants depends on point For 33 the point is in quadrant 1 so 9 g For 33 the point is in quadrant 4 so 9 57 Ex 1 Convert from polar form to rectangular form apoint25n r22 9 5 T l x rc059 2c053 22 1 yrsin925in5n2 T 1 b curve 1quot 3 c059 i c059 2 r 37x 1 2 3x y2 x2 2 3x circle ii r23rcosl9 xrcosl9 x2y23x 2 Convert from rectangular form to polar form a point 1 V3 TWV1 32 tan9 3 92 35 b curve 2x 3y 2 4 2rcosl93rsin94 r 2cosQ351n0 Graphs of polar equations Attached Handout For slope of a tangent in regards to polar equations Given polar curve 1 f 9 What is 2 xrcosl9 y r Sin 9 This is a parametric equation so d d dy a f 6 sin0f0 cos 0 d 81n0rcos0 dx f 0 cos 0 fX sin 9 c030 1 sin 0 Area under a curve region In rectangular form the representation for area under a curve is f f x gxdx Where the area is incremented by Widths of rectangles In polar form the area is incremented by s arc length area 0 sector 6 6 1 4 area of sector 2 gtlt7tr2 r26 radians areaof ClTCle 2n 2n 2 Consider a polar curve 7 f 9 The interval of integration is from 9 a to 9 b R is the total region bounded by r f 9 AA is the area of the incremented sectors of 39 u 39 39 i H the curve u Therefore AA fx2A9 so A f69fx2 d9 EX 1 Find the area of one pedal of r cos59 Need to nd out the range of integration Find Where r 0 When 59 E 9 l 2 10 The rst pedal is then bound in the region 110110 1 2 1 721 1 1 quot10 soA 2 XEfO cos 56MB f0 E1c05106d9 9 1 Osm106 ea 01 2 Find the area of the region bounded by the graphs r 3 2 cos 9 and r 3 sin 9 O a nd their intersection points 3 2cosB3 sin9 tan91 6 and b Find Which is the upper function and Which is the lower function plot some points in the limits of integration V r3 Zsin9 r3 2cosl9 r 0 3 o 1i 0 9 1 W2 3 nz 3 7t 5 n 5 37T2 3 ant2 3 2 1 2n AA farthest sector from pole shortest sector from pole AA 3 2 cos 92A9 3 2 sin 92A9 5 1 T A 22 9 12c0594c0529 3 125in64sin29 Z From here it becomes a long but doable integration problem remember the power reducing formulas when doing this

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