Question

4. Consider the area of the region that lies inside the curve given in polar form) by r = 6 sin(@) and outside the cardioid given by r=2+2 sin(0). (a) (3pts) Set up but do not evaluate an integral(s) which represents the area of this region. (b) (3.5pts) Evaluate this integral to determine the exact area of this region. (Hint: you will need to use a trig, identity)

Answer 1

Ans: ai Given r = 6 Sin (O) and r=2+2 sin(0) To determine this wrea we will need to know the valves 0 the two curves intersect. we Can for which delermine equalions and points by setting the these solving two 6 Sind = at asino 4 Sind Sino / T 6 T - T ) 11 G ST 6 Hence 57% 1 2 [ (s Sino)? - (2 +2510) ?] do 46 اط A = 1 SSM (365ip?o 36 Sino – 4-45420 - 8 Sino) d 2 TY 546 (16 sine Sino -2) do TY A = & Sino - 2 ( [0 / 9-420) -2) de 5% (-8C828 – 4 sino +6) de A N 5746 4 Sin20 + 4 Coso + 609

A = -4 sin (511) + 4 ose + + 4COSTI T 6 (571) - {-4 sin 4e-+ 4x(-9) +51 - -{ -4% A = - 4 x 3 2 + 4x Ako + ] 2 A = 413 4 B 2 + 5T + 4 3 2 Yon T Aa 5T ㅈ A = 40

5 3 2 - 55 4 3 自