Question

Show all work please! 1. Graph the polar equations r = 3-2sin(0) and r = 2. Find and label any key points that you will need to find the area that lies in the common interior of the two polar curves. Set up the integral(s) and please show all the work to integrate and evaluate your integral manually.

Answer 1

Solve. The graph of the polar equations y = 3- 2 sinlo) 8=2 shown below. The region common to both curres in o ABC and 8 - 2 А c 7 = 3-2 sinio) To determine the area, we shall need to know the values of o for which the two curves interset. he can determine these points by setting the two equations and solving re 3-2 sin (0) = 2

- 2 sincol= -1 F%. sin (0)= / SA Here is the sketch of the figure with these ongles. = 2 0= 57% area outsice 8=3-2.sin(e) and inside T/ B Now area of OA BC : larea of circle 8= 2) - (ureu outsiche >=3-2 sinto) and in 8=2)

Here area outside the Cuove Y = 3-2 sinlos and inside r=2 is 5716 Ź (12) 716 ( 3-2 sin(o))2) do 03(2-69 vedia 4- (9+4 sin218) – 12 sinio)) do 716 5716 -5 - 4 pin?(0) + 12 pinlol) do alo 5746 11 Ź (5+12 +12 1in(0) + 2 cos(20) -2) do 17 546 1 2 -7 + 12 sin(a) + 2 los (20) do *16

악 5지 -7-12 세이 +din (20) (to) 2. 17 2 ㅈ 3 This area of the common interior of the two polar Curres is TT (2) 2 - 1165 AT- | 17 + ㅈ 2 3 1. 19즈 3 - 10.370 2.
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