The polar curves $r=1-\cos{\theta}$ and $r=1+\cos{\theta}$ are shown in the figure above. Which of the following expressions gives the total area of the shaded regions?

Summary Submit

The formula for area bounded by the graph is:

$$\frac{1}{2}\int \limits_a^b (r(\theta))^2   d\theta$$

If we can obtain the area for one of the four quadrants, we can simply quadruple that value.

The following integral provides the area for the top right quadrant of the cardioid ($r=1-\cos{\theta}$):

$$\frac{1}{2}\int\limits_{0}^{\pi/2} (1-\cos{\theta})^2   d\theta$$

Quadrupling this value results in the answer:

$$4\cdot \frac{1}{2}\int\limits_{0}^{\pi/2} (1-\cos{\theta})^2   d\theta = \boxed{2\int\limits_{0}^{\pi/2} (1-\cos{\theta})^2   d\theta}$$

The other answer choices represent other areas (like the larger regions) or do not have the right coefficient to reflect additional regions.