Questions: Find the area of the region inside (r=5 sin theta) but outside (r=4).

Solution Steps

To find the area of the region inside \( r = 5 \sin \theta \) but outside \( r = 4 \), we need to set up and evaluate the appropriate integral in polar coordinates. The area can be found by integrating the difference of the squares of the radii over the appropriate range of \(\theta\).

We are given two polar curves: \( r = 5 \sin \theta \) and \( r = 4 \). The area we want to find is the region that lies inside the curve \( r = 5 \sin \theta \) and outside the circle \( r = 4 \).

To find the points where the two curves intersect, we set them equal to each other: \[ 5 \sin \theta = 4 \] This gives us: \[ \sin \theta = \frac{4}{5} \] The solutions for \( \theta \) are: \[ \theta_1 = \arcsin\left(\frac{4}{5}\right) \approx 0.9273 \] \[ \theta_2 = \pi - \theta_1 \approx 2.2143 \]

The area \( A \) of the region can be calculated using the formula: \[ A = \frac{1}{2} \int_{\theta_1}^{\theta_2} \left( (5 \sin \theta)^2 - (4)^2 \right) d\theta \] This integral represents the area between the two curves from \( \theta_1 \) to \( \theta_2 \).

After evaluating the integral, we find that the area of the region is approximately: \[ A \approx 3.7477 \]

Final Answer