could the points of integration be 2pi/3 and 4pi/3?

Yes, you're correct. The points of integration for the inner loop of the polar curve \( r = 2 + 4 \cos \theta \) should be \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\). Let's re-evaluate the steps with these points:

1. **Find the points where \( r = 0 \)**:
   
   \[
   2 + 4 \cos \theta = 0 \implies \cos \theta = -\frac{1}{2}
   \]

   \[
   \theta = \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{3}
   \]

2. **Set up the integral**: The area of a polar region is given by

   \[
   A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta
   \]

   For the inner loop, we integrate between \(\theta = \frac{2\pi}{3}\) and \(\theta = \frac{4\pi}{3}\):

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

Therefore, the integral that computes the area of the inner loop is:

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

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you derive the formula for the area of a polar region?
2. What is the significance of finding the intersection points in polar coordinates?
3. How do you determine the limits of integration for a polar curve?
4. How does the symmetry of the curve affect the setup of the integral?
5. What are the general steps to set up an integral for a polar curve?
6. How do you interpret the result of a polar area integral geometrically?
7. Can you provide an example of a similar polar curve and its area integral?
8. How would you handle the integral if the curve had no inner loop?

**Tip:** When working with polar coordinates, always verify the points of intersection by checking the values of \( r \) at those points to ensure correct integration limits.

Learn how to calculate the area of the inner loop in the polar curve r = 2 + 4 cos(theta). Understand the integration steps and the significance of the points of integration (2pi/3 and 4pi/3) in polar coordinates. Suitable for advanced high school or college-level students.

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