Chapter 13.3, Problem 49E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a.    Let R be the unit disk centered at (0,0). Then

     ${\displaystyle {\iint }_R\left(x^2+y^2\right)dA={\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^1{r}^{2}drd\theta }}}.$

b.    The average distance between the points of the hemisphere $z=\sqrt{4-x^2-y^2}$ and the origin is 2 (calculus not required).

c.    The integral ${\displaystyle {\int }_0^1{\displaystyle {\int }_0^{\sqrt{1-y^2}}{e}^{x^2+y^2}d}xdy}$ is easier to evaluate in polar coordinates than in Cartesian coordinates.