Use iterated integral to find the area enclosed by r=2sin2(theta).

Question

How do i graph this?

Answer 1

For the graph, see

http://www.wolframalpha.com/input/?i=+r+%3D+2sin%5E2%CE%B8

For the area, recall that in polar coordinates,

a = ∫∫ r dr dθ
= ∫[0,2π] ∫[0,2sin^2θ] r dr dθ
= ∫[0,2π] 1/2 r^2 dθ [[0,2sin^2θ]
= ∫[0,2π] 1/2 (2sin^2θ)^2 dθ
= ∫[0,2π] 2 sin^4θ dθ
= 3π/2

You have to use integration by parts twice, so I'll leave the details to you.