Find the area of the region that lies inside both curves

Question

r=7sin(2θ) r=7sin(θ)

Answer 1

The curves are r = 7sin(2θ) and r = 7sin(θ)

First we find out the point where the two curves intersect.So equate the the two curve.

7sin(2θ) = 7sin(θ)

2sinθ cosθ = sinθ

2cosθ = 1

cosθ = 1/2

θ = π/3.

Graph:

Draw the graph in polar - cordinate plane

We can observe from the graph that the two curves intersect at θ = π/3 line.

Now first we find the area of the bounded region in first then we double it for the total area.

The bounded region is divide into two region(as highligted in the graph)

One of the region, the θ varies from 0 to π/3 and the curve is r = 7sin(θ)

Second region, the θ varies from π/3 to π/2 and the curve is r = 7sin(2θ)

Area of the bounded is given by

We know that and 

Therefore area of the bounded region is .

Comments

You then need to multiply the answer by 2 because there are two areas covered. Easy to forget.