The graph in the image represents a polar curve given by the equation $r = 2 \cos(2\theta)$. The problem asks to find the area of the region $D$, which is one of the four petals of the curve, using double integrals.

Step-by-step solution:

1. Understanding the equation: The polar equation $r = 2 \cos(2\theta)$ describes a four-petaled rose curve (also called a "quadrifolium"). Each petal corresponds to a specific range of $\theta$.

2. Setting up the integral: The area $A$ of one petal (the $D$ region) can be computed using the formula for the area in polar coordinates: $A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 \, d\theta$ where $r = 2 \cos(2\theta)$, and $\theta_1$ and $\theta_2$ are the bounds for the angle that describe one petal.

3. Determining the limits of integration: The rose curve repeats every $\pi$ radians due to the factor of $2\theta$. To find the bounds for one petal, observe that $r = 0$ when $\theta = \frac{\pi}{4}$ and $\theta = 0$. So, we can integrate from $0$ to $\frac{\pi}{4}$ to capture one petal.

4. Setting up the integral for one petal: Substituting $r = 2 \cos(2\theta)$ into the formula for the area: $A = \frac{1}{2} \int_0^{\frac{\pi}{2}} \left(2 \cos(2\theta)\right)^2 d\theta$ Simplifying the expression: $A = 2 \int_0^{\frac{\pi}{2}} \cos^2(2\theta) \, d\theta$

5. Using a trigonometric identity: To integrate $\cos^2(2\theta)$, use the identity: $\cos^2 x = \frac{1 + \cos(2x)}{2}$ Applying this to the integral: $A = 2 \int_0^{\frac{\pi}{2}} \frac{1 + \cos(4\theta)}{2} \, d\theta$ $A = \int_0^{\frac{\pi}{2}} (1 + \cos(4\theta)) \, d\theta$

6. Evaluating the integral: $A = \left[\theta + \frac{\sin(4\theta)}{4}\right]_0^{\frac{\pi}{2}}$ At the bounds $\theta = 0$ and $\theta = \frac{\pi}{2}$, the sine term vanishes, so the result simplifies to: $A = \frac{\pi}{2}$

Thus, the area of one petal $D$ is $\frac{\pi}{2}$.

Would you like further details or clarifications on any part of the process? Here are some related questions you may find useful:

1. How do you calculate the area for multiple petals in polar coordinates?
2. Why does the polar curve equation have four petals?
3. What are other common forms of rose curves, and how do you calculate their areas?
4. How does the trigonometric identity simplify the integration process?
5. Can this method be extended to curves with different symmetries or petal counts?

Tip: Always remember to use the correct limits for the angle $\theta$ when dealing with polar integrals to avoid miscalculating areas.