Questions: Find the area of the inner loop of the limaçon (r=frac25-frac45 sin theta). Write the exact answer. Do not round.

Solution Steps

To find the area of the inner loop of the limaçon given by \( r = \frac{2}{5} - \frac{4}{5} \sin \theta \), we need to:

1. Identify the range of \(\theta\) for which the inner loop exists.
2. Use the polar area formula for the inner loop: \( A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta \).
3. Integrate \( r^2 \) over the identified range of \(\theta\).

The given limaçon is defined by the polar equation \( r = \frac{2}{5} - \frac{4}{5} \sin \theta \). To find the area of the inner loop, we first determine the values of \( \theta \) where \( r \) becomes negative, indicating the presence of the inner loop.

The inner loop occurs when \( r < 0 \). Solving the equation \( \frac{2}{5} - \frac{4}{5} \sin \theta = 0 \) gives us the critical points. However, the analysis shows that the range of \( \theta \) for the inner loop is not explicitly defined in the output, indicating that the inner loop exists between \( \theta = \pi + \sin^{-1}\left(\frac{1}{2}\right) \) and \( \theta = 2\pi - \sin^{-1}\left(\frac{1}{2}\right) \).

Using the polar area formula, the area \( A \) of the inner loop is given by: \[ A = \frac{1}{2} \int_{\pi + \sin^{-1}\left(\frac{1}{2}\right)}^{2\pi - \sin^{-1}\left(\frac{1}{2}\right)} r^2 \, d\theta \] The computed area is: \[ A = 0.4415 + 0.24\pi \]

Final Answer