Given the polar curves $r=7 \sin \left(\theta \right)$ and $r=7 \cos \left(\theta \right)$, what is the area of the region that lies inside both curves?

• A. $\frac{49\pi }{4}$
• B. $\frac{49\pi }{8}$
• C. $49$
• D. $\frac{49\pi }{2}$

Answer: (B)

Verified step by step guidance

1. Identify the given polar curves: \(r = 7 \sin(\theta)\) and \(r = 7 \cos(\theta)\). These represent circles in polar coordinates.
2. Find the points of intersection by setting \$7 \sin(\theta) = 7 \cos(\theta)\(, which simplifies to \)\sin(\theta) = \cos(\theta)\(. Solve for \)\theta$ to find the limits of integration for the overlapping region.
3. Determine the area inside both curves by integrating the minimum of the two \(r^2\) values over the range of \(\theta\) where they overlap. The formula for the area enclosed by a polar curve is \(\frac{1}{2} \int r^2 \, d\theta\).
4. Set up the integral for the area of the overlapping region as the sum of two integrals: one for \(r = 7 \sin(\theta)\) from \(\theta = 0\) to the intersection angle, and one for \(r = 7 \cos(\theta)\) from the intersection angle to \(\theta = \frac{\pi}{2}\).
5. Evaluate the integrals (without calculating the final numeric value here) and add the results to find the total area inside both curves.