Calculating the area between two functions expressed in polar coördinates

Question

I have the following to polar coördinates: $r=1+\cos(\theta)$ and $r=3\cos(\theta)$.

The question is to calculate the area in side $r=1+\cos(\theta)$ and outside $r=3\cos(\theta)$. I know I need to use the formula $$\int\frac{1}{2}r^2d\theta$$ But I don't really know which boundaries to choose for both polar coördinates.

Answer 1

Sometimes one function is outside, sometimes the other. How do you want to integrate such a region?

Answer 2

When you draw a graph you will notice that $r=1+\cos\theta$ is a cardioid and $r=3\cos\theta$ is a circle.

To find the area inside $r=1+\cos\theta$ and outside $r=3\cos\theta$ you need to split the integral into two parts.

You also need to find out the intersection point of those two areas.

So, to find that we need $$1+r\cos\theta=3\cos\theta$$ $$\theta=\dfrac{\pi}{3},\dfrac{5\pi}{3}$$ So, the Area we need is $$A=\int_{\frac{\pi}3}^{\frac{\pi}{2}}\int_{3\cos\theta}^{1+\cos\theta}r\ drd\theta=1-\dfrac{\pi}{4} $$