Question

Use a double integral in polar coordinates to find the area of the region common to the interior of the cardioids

r = 1 + cos θ and r = 1 − cos θ.

Answer 1

In order to calculate the area between the curves, we determine the points of intersection:

$$1+\cos \theta=1-\cos \theta$$

$$\cos \theta=0$$

$$\theta=\pi/2$$

$$\theta=-\pi/2$$

The area is calculated using the following equation:

$$A=\displaystyle\int_{-\pi/2}^{\pi/2}\displaystyle\int_{1-\cos \theta}^{1+\cos \theta}rdrd\theta$$

Using symmetry:

$$A=\displaystyle\int_{-\pi/2}^{\pi/2}\displaystyle\int_{1-\cos \theta}^{1+\cos \theta}rdrd\theta=2\displaystyle\int_{0}^{\pi/2}\displaystyle\int_{1-\cos \theta}^{1+\cos \theta}rdrd\theta$$

$$=2\displaystyle\int_{0}^{\pi/2}\bigg[\dfrac{r^2}{2}\bigg]\begin{matrix} 1+\cos \theta \\ 1-\cos\theta \end{matrix}d\theta$$

$$=2\displaystyle\int_{0}^{\pi/2}(4\cos\theta)d\theta=8[\sin \theta]\begin{matrix} \pi/2 \\ 0 \end{matrix}=8({units}^2)$$