Given the polar curves $r=5\sin \left(\theta \right)$ and $r=5\cos \left(\theta \right)$, what is the area of the region that lies inside both curves?

• A. $25$
• B. $\frac{25}{2}$
• C. $\frac{25\pi }{4}$
• D. $\frac{25\pi }{8}$

Answer: (D)

Verified step by step guidance

1. Identify the given polar curves: \(r = 5 \sin(\theta)\) and \(r = 5 \cos(\theta)\). These represent circles in polar coordinates.
2. Find the points of intersection by setting the two equations equal: \$5 \sin(\theta) = 5 \cos(\theta)\(, which simplifies to \)\sin(\theta) = \cos(\theta)\(. Solve for \)\theta$ to find the limits of integration.
3. Determine the region inside both curves. Since both are circles, the overlapping region is symmetric and bounded between the angles found in the previous step.
4. Set up the integral for the area of the intersection. The area inside a polar curve from \(\alpha\) to \(\beta\) is given by \(\frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta\). For the overlapping region, the total area is the sum of the areas under each curve over their respective intervals where one is inside the other.
5. Evaluate the integrals separately for each curve over the appropriate limits and add them to find the total overlapping area. Remember to square the \(r\) values inside the integral, i.e., use \(r^2 = 25 \sin^2(\theta)\) or \$25 \cos^2(\theta)$.