Chapter 5.3, Problem 147E

In the following exercises, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates.

147. ${\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{x^2}{\overset{x}{\int }}\frac{y}{\sqrt{x^2+y^2}}dydx={\displaystyle \underset{0}{\overset{\pi /4\tan \theta }{\int }}{\displaystyle \underset{0}{\overset{\sec \theta }{\int }}r\sin \theta drd\theta }}}}$