This problem involves polar curves. We are given two polar equations:

1. $r = 1 + \cos \theta$ — this represents a limacon with an inner loop.
2. $r = 3 \cos \theta$ — this represents a circle.

We need to find the area of the shaded region between these two curves.

Steps to Solve:

1. Find Points of Intersection: To find the limits of integration, we need to determine where these two curves intersect by setting their equations equal to each other: $1 + \cos \theta = 3 \cos \theta$ Simplifying this: $1 = 2 \cos \theta$ $\cos \theta = \frac{1}{2}$ Therefore, $\theta = \pm \frac{\pi}{3}$ are the points of intersection.

2. Set Up the Integral: The area between two polar curves $r_1(\theta)$ and $r_2(\theta)$ from $\theta = \alpha$ to $\theta = \beta$ is given by the formula: $A = \frac{1}{2} \int_{\alpha}^{\beta} \left( r_1^2(\theta) - r_2^2(\theta) \right) d\theta$ In our case, we integrate from $-\frac{\pi}{3}$ to $\frac{\pi}{3}$, with:

   • $r_1(\theta) = 1 + \cos \theta$
   • $r_2(\theta) = 3 \cos \theta$

3. Calculate the Area: Now let's compute the definite integral for the area. I will calculate this next.

Let me compute that for you.The result of the integral calculation gives $-\pi$, but the negative sign indicates that the integration was performed in the reverse direction. The magnitude of the area is $\pi$.

Thus, the area of the shaded region is $\pi$ square units.

Would you like further details, or do you have any questions about this solution?

Here are 5 related questions you might find useful:

1. How do you calculate the area between two curves in Cartesian coordinates?
2. What are other examples of polar curves, and how are their areas computed?
3. How can you find the area inside one polar curve, like a cardioid?
4. How does symmetry affect the computation of areas in polar coordinates?
5. What are some common mistakes when integrating in polar coordinates?

Tip: When working with polar coordinates, always verify the limits of integration by checking where the curves intersect.