5. Consider the integral $\int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}}\left(x^{2}+y^{2}\right)^{5 / 2} d x d y$.
(a) Sketch the region of integration for this integral. Describe it in polar coordinates.
(b) Convert the integral to polar coordinates and evaluate it.
Answer
The final answer is \(\boxed{\frac{1}{18}\pi}\).
Step 1 ：The region of integration for the integral \(\int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}}(x^{2}+y^{2})^{5 / 2} dx dy\) is a quarter of a unit disk in the first quadrant. In polar coordinates, this region is described by \(0 \leq r \leq 1\) and \(0 \leq \theta \leq \frac{\pi}{2}\).
Step 2 ：The integral in polar coordinates is given by \(\int_{0}^{\pi/2} \int_{0}^{1} r^{2}(r^{2})^{5/2} r dr d\theta\).
Step 3 ：The integrand is \(r^{3}(r^{2})^{2.5}\).
Step 4 ：The integral evaluates to \(0.0555555555555556\pi\).
Step 5 ：The final answer is \(\boxed{\frac{1}{18}\pi}\).
