Find the area of the region common to the cardioids r = 12 + 12sin() and r = 12 - 12sin().

Round your answer to four decimal places.

To find the area of the region common to the two cardioids \( r = 12 + 12\sin\theta \) and \( r = 12 - 12\sin\theta \), we need to follow these steps:

### 1. Sketch the curves and identify the common region
The given equations are:
- \( r_1 = 12 + 12\sin\theta \)
- \( r_2 = 12 - 12\sin\theta \)

These are cardioids, and the common region is the area that lies inside both curves.

### 2. Find the points of intersection
To find the points of intersection, set the two equations equal to each other:
\[
12 + 12\sin\theta = 12 - 12\sin\theta
\]
\[
24\sin\theta = 0
\]
\[
\sin\theta = 0
\]
The solutions are \( \theta = 0 \) and \( \theta = \pi \).

### 3. Set up the integral to find the common area
The common area between the curves can be found using the formula:
\[
A = \frac{1}{2} \int_{\theta_1}^{\theta_2} \left(r_1^2 - r_2^2\right) \, d\theta
\]
Where \( r_1 = 12 + 12\sin\theta \) and \( r_2 = 12 - 12\sin\theta \), and \( \theta_1 = 0 \) and \( \theta_2 = \pi \).

### 4. Evaluate the integral
First, compute \( r_1^2 \) and \( r_2^2 \):
\[
r_1^2 = (12 + 12\sin\theta)^2 = 144 + 288\sin\theta + 144\sin^2\theta
\]
\[
r_2^2 = (12 - 12\sin\theta)^2 = 144 - 288\sin\theta + 144\sin^2\theta
\]
Subtracting:
\[
r_1^2 - r_2^2 = (144 + 288\sin\theta + 144\sin^2\theta) - (144 - 288\sin\theta + 144\sin^2\theta) = 576\sin\theta
\]
So the area is:
\[
A = \frac{1}{2} \int_{0}^{\pi} 576\sin\theta \, d\theta
\]
\[
A = 288 \int_{0}^{\pi} \sin\theta \, d\theta
\]
The integral of \( \sin\theta \) is \( -\cos\theta \), so:
\[
A = 288 \left[-\cos\theta \right]_{0}^{\pi}
\]
\[
A = 288 \left[-\cos(\pi) + \cos(0)\right]
\]
\[
A = 288 \left[-(-1) + 1\right] = 288 \times 2 = 576
\]

### 5. Final Answer
The area of the region common to both cardioids is \( \boxed{576} \) square units.

If you'd like any more details or have questions, let me know!

Here are some related questions you could explore:
1. What are the Cartesian forms of the given cardioids?
2. How does changing the coefficients in a cardioid equation affect its shape?
3. Can you find the area enclosed by each cardioid individually?
4. What are the general properties of a cardioid?
5. How would the area change if the cardioids were shifted or rotated?
6. Can you find the length of the boundary of the common region?
7. How would the problem change if we used polar equations of circles instead of cardioids?
8. What is the significance of the points where the cardioids intersect?

**Tip:** When dealing with polar coordinates, it's useful to visualize the curves to better understand the regions of integration.

Explore how to find the area of the region common to two cardioids r = 12 + 12sin(θ) and r = 12 - 12sin(θ). Follow a step-by-step solution involving polar coordinates, integration, and trigonometric calculations. Suitable for advanced high school students and beyond.

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