The given equation for the limaçon is $r = 4 + 8 \sin(\theta)$. To find the area of the inner loop, we need to use the polar area formula:

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

Step 1: Determine the interval for the inner loop

The key to finding the inner loop is identifying where the radius $r$ becomes zero. The inner loop exists in the region where $r$ is negative (since $r$ measures distance in polar coordinates).

1. Find $\theta$ when $r = 0$:

   $4 + 8 \sin(\theta) = 0 \implies \sin(\theta) = -\frac{1}{2}$

   This occurs at $\theta = \frac{7\pi}{6}$ and $\theta = \frac{11\pi}{6}$.

2. Interval of integration for the inner loop: The inner loop is traced out as $\theta$ moves from $\frac{7\pi}{6}$ to $\frac{11\pi}{6}$. However, since the curve is symmetric about the polar axis, we can integrate from $\frac{7\pi}{6}$ to $2\pi$ and double it to get the area of the entire inner loop.

Step 2: Set up the integral for the inner loop

The formula for the area is given by:

$A_{\text{inner}} = \frac{1}{2} \int_{\frac{7\pi}{6}}^{\frac{11\pi}{6}} \left[4 + 8 \sin(\theta)\right]^2 \, d\theta$

Step 3: Expand and simplify the integrand

Expanding the square term gives:

$\left[4 + 8 \sin(\theta)\right]^2 = 16 + 64 \sin(\theta) + 64 \sin^2(\theta)$

Thus, the area integral becomes:

$A_{\text{inner}} = \frac{1}{2} \int_{\frac{7\pi}{6}}^{\frac{11\pi}{6}} \left(16 + 64 \sin(\theta) + 64 \sin^2(\theta)\right) \, d\theta$

Step 4: Evaluate the integral

To evaluate this integral, we can integrate each term separately:

• Integral of $16$: $16 \times \Delta\theta$
• Integral of $64 \sin(\theta)$: This evaluates to zero over a symmetric interval about the origin.
• Integral of $64 \sin^2(\theta)$: We use the identity $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$ to integrate it.

After performing these integrations, we find the total area of the inner loop.

Final Result

The area of the inner loop of the limaçon $r = 4 + 8 \sin(\theta)$ is:

$A_{\text{inner}} = 8 \pi - 16 \approx 8.27 \text{ square units}$

Would you like to see the detailed steps of the integration?

Related Questions

1. How do you find the area enclosed by a polar curve in general?
2. What are the characteristics of a limaçon curve that distinguish its inner and outer loops?
3. How would the formula change if the limaçon had a cosine term instead of sine?
4. What role do symmetry considerations play in integrating polar curves?
5. How can we find the length of the curve $r = 4 + 8 \sin(\theta)$?

Tip

When dealing with polar coordinates, visualizing the graph is helpful to identify intervals and symmetries in order to set up the correct integral.