Find the area of the region that is outside the curve $r=1 – \cos\theta$ but inside the curve $r=1$.
Drawing the graphs, one way to get the desired area is to first obtain the area of the semicircle ($\pi/2$) and subtract from it $2\times$ the integral from $0$ to $\pi/4$ for the curve $r=1-\cos\theta$. Therefore, I set up the equation to be $\frac{\pi}{2} – 2\cdot0.5\cdot \int_{0}^{\pi/4} (1-\cos(\theta))^2 \,d\theta = \frac{\pi}{8} + \sqrt{2} – \frac{1}{4}$. However, this is not the right answer. Where might I have gone wrong?
Best Answer
The area that you are interested in is the area of the region outside the blue cardioid and inside the red circle in the picture below:
The area of the part of the cardioid located in the first and fourth quadrants is$$\int_{-\frac\pi2}^\frac\pi2(1-\cos\theta)^2\,\mathrm d\theta=\frac{3\pi}2-4.$$So, the area that your after is equal to$$\frac\pi2-\left(\frac{3\pi}2-4\right)=4-\pi.$$