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We are tasked with finding the area common to the interiors of the cardioids \( r = 1 - \sin \theta \) and \( r = 1 + \cos \theta \).

### Steps to solve this:

1. **Find the intersection points** of the two curves by setting their equations equal to each other:

   \[
   1 - \sin \theta = 1 + \cos \theta
   \]

   Simplifying this:

   \[
   - \sin \theta = \cos \theta \implies \tan \theta = -1
   \]

   The solutions to \( \tan \theta = -1 \) are \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{7\pi}{4} \).

2. **Set up the integral** to find the area common to the two curves. The area common to the cardioids is given by:

   \[
   A = \frac{1}{2} \int_{\theta_1}^{\theta_2} \left( r_{\text{outer}}^2 - r_{\text{inner}}^2 \right) d\theta
   \]

   Here, \( r_{\text{outer}} \) is the cardioid with the larger value at a given \( \theta \). After analyzing the two curves, \( r = 1 + \cos \theta \) serves as the outer curve for \( \theta \) between \( 0 \) and \( \pi \), and \( r = 1 - \sin \theta \) serves as the outer curve for \( \theta \) between \( \pi \) and \( 2\pi \).

   Therefore, the total area is split into two parts:

   \[
   A = \frac{1}{2} \int_{0}^{\frac{3\pi}{4}} (1 + \cos \theta)^2 d\theta + \frac{1}{2} \int_{\frac{3\pi}{4}}^{2\pi} (1 - \sin \theta)^2 d\theta
   \]

3. **Evaluate the integrals**:

   Now, let's compute the two integrals to find the area.

I'll compute this for you.The area of the region common to the interiors of the two cardioids \( r = 1 - \sin \theta \) and \( r = 1 + \cos \theta \) is:

\[
A = 0.75 + \sqrt{2} + \frac{3\pi}{2}
\]

This is the exact expression for the area.

Would you like further details on any steps? Here are some related questions:

1. How do you compute the area enclosed by a single cardioid?
2. What are the geometric properties of a cardioid in polar coordinates?
3. Can you explain how to set up integrals for polar regions more generally?
4. How does the symmetry of cardioids affect the integration limits?
5. How would the area change if the curves were shifted or scaled?

**Tip**: When working with polar curves, symmetry often simplifies the integration process, especially for cardioids and circles.

Find the area of the region common to the interiors of the cardioids r = 1 - sin(θ) and r = 1 + cos(θ).

This problem demonstrates how to find the area common to the interiors of two cardioids in polar coordinates. It involves calculating the intersection points and setting up integrals to compute the exact area. Ideal for undergraduate students learning calculus and polar coordinates.

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