Set up the iterated double integral that represents the area of the region enclosed by both the cardioids r = 1 + cos(θ) and r = 1 - cos(θ).

Steps to Solve

1. Identify the Region of Intersection

To find the area enclosed by the two cardioids, we first need to determine where they intersect. Set their equations equal to each other: $$ 1 + \cos(\theta) = 1 - \cos(\theta) $$

2. Solve for Intersection Points

By simplifying the equation: $$ \cos(\theta) + \cos(\theta) = 0 $$ $$ 2\cos(\theta) = 0 $$ This gives us: $$ \cos(\theta) = 0 $$ The solutions are: $$ \theta = \frac{\pi}{2}, \frac{3\pi}{2} $$

3. Set the Limits of Integration

The area can be found using polar coordinates. The region enclosed by the cardioids will be between these angles. The area can be expressed as: $$ A = \frac{1}{2} \int_{\theta_1}^{\theta_2} \left( \text{outer radius}^2 - \text{inner radius}^2 \right) d\theta $$

Here, the outer and inner radii depend on the angle:

• For $\theta \in [0, \frac{\pi}{2}]$, outer radius $r = 1 + \cos(\theta)$ and inner radius $r = 1 - \cos(\theta)$.
• The area can be calculated from $\theta = 0$ to $\theta = \frac{\pi}{2}$, and then multiplied by 4 to cover the entire area.

4. Express the Integral

The iterated double integral for the area is: $$ A = 2 \int_{0}^{\frac{\pi}{2}} \left( (1 + \cos(\theta))^2 - (1 - \cos(\theta))^2 \right) d\theta $$