Polar coordinate calculation for area bounded by $r=2(1+\cos \theta)$ and $r=2\cos \theta$.

Question

Calculate the area of the region that is bounded by $r=2(1+\cos \theta)$ and $r=2\cos \theta$.

I solved and found $4\pi$ but the answer is written $5\pi$.

enter My Solution

Answer 1

The function $\cos\theta$ alone finishes one complete turn when $\theta=\pi$. On the other hand, $1+\cos\theta$ is as you have expected it to be. You should've looked for when both curves reach $0$ for the first time and then proceeded.

Answer 2

Note that it takes $\theta$ to vary from $-\pi$ to $\pi$ for the curve $r=2(1+\cos \theta)$ to trace a complete loop, while it takes $\theta$ to vary from $-\pi/2$ to $\pi/2$ for the curve $r=2\cos \theta$ to trace a complete loop.

Thus, the area between them is given by

$$\int_{-\pi}^{\pi} \frac12 [2(1+\cos \theta)]^2d\theta + \int_{-\pi/2}^{\pi/2} \frac12 (2\cos \theta)^2d\theta =6\pi-\pi=5\pi$$