# Calculating the area between two functions expressed in polar coördinates

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.

• The tag says "polar-coordinates" (not polar coördinates:) ) – Dietrich Burde Jun 5 at 18:51

Sometimes one function is outside, sometimes the other. How do you want to integrate such a region? 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 $$A=\int_{\frac{\pi}3}^{\frac{\pi}{2}}\int_{3\cos\theta}^{1+\cos\theta}r\ drd\theta=1-\dfrac{\pi}{4}$$