Area inside a curve and outside a Cardoid

Question

For a National Board Exam:

Find the area which is inside the curve r=3cos(theta) and outside the cardoid r=1+cos(theta)

Answer is pi

Ok I am trying to setup the right definite integral for the calculator, I've tried:

Using the formula for finding area of polar curves: $${ A = \int^b_a \frac{1}{2} f(\theta)^2 d\theta}$$

$${ A = \int^{2\pi}_0 \frac{1}{2} ( 3cos(\theta) - (1+cos(\theta)))^2 d\theta = 3\pi}$$

What is the right integral?

Answer 1

HINT...You need to establish where the curves intersect in order to determine the limits, so solve $3\cos \theta=1+\cos \theta$.

So your limits are $\pm\frac {\pi}{3}$

Answer 2

First figure out where the two curves intersect:

$$3\cos{\theta}=1+\cos{\theta} \\ \cos{\theta}=\frac{1}{2} \\ \theta=\frac{\pi}{3},\, \frac{5\pi}{3}$$

Then evaluate the areas inside the two curves and subtract, using these intersection angles as the bounds.