Find the area of the specified region Inside one leaf of the

Find the area of the specified region. Inside one leaf of the four - leaved rose r = 7 sin 2 theta

Solution

Area of R= sin(2 theta) is Formula for finding area using polar equations A = ?[?=a..b] 1/2 r² d? So we need to integrate from ?=0 to ?=2p to get area of full four-leaf clover A = 1/2 ?[?=0..2p] sin²(2?) d? Now it\'s difficult to integrate sin²(2?), so we use half angle formula sin²(a) = 1/2 (1 - cos(2a)), where a = 2? sin²(2?) = 1/2 (1 - cos(4?)) A = 1/2 ?[?=0..2p] 1/2 (1 - cos(4?)) d? A = 1/4 ?[?=0..2p] (1 - cos(4?)) d? A = 1/4 (? - sin(4?)/4) |[?=0..2p] A = 1/4 ( (2p-sin(8p)) - (0-sin(0)) ) A = p/2 Now, for r= 7 sin(2 theta) area is 49* pi/2 but area of one leaf is 49*pi/2*4 = 49 pi/8