computing areas and average values

Question

1. Sketch each region and use a double integral to find its area:

The region inside both the cardioid r=1-cosθ and the circle r=1

2. Find the following average values

The average distance between points within the cardioid r=1+cosθ and the origin.

Thank you

Answer 1

a. By "area inside both" if you means "intersection" of the circle r=1 and cardiod r=1-cosθ, then you can calculate the area, among others, the following way.

First, area is calculated as A=∫(1/2)f²(θ)dθ.

Now, we find where the two curves intersect.

1-cos(θ)=1.

θ=-Pi/2 and Pi/2

So, the area is

A=(1/2)∫_-π/2^π/2(1- (1-cos²(θ))2)dθ=..=2-π/4.

b. In fact, the equation r=1+cosθ itself shows the distance from origin to the each point of the cardioid. The average distance would be 2π.