Chapter 14.6, Problem 25ES

(a) Use a CAS to evaluate ${\displaystyle {\int }_{-2}^2{\displaystyle {\int }_1^4{\displaystyle {\int }_{\pi /6}^{\pi /3}\frac{r{\tan }^3\theta }{\sqrt{1+z^2}}d\theta drdz}}}$

(b) Find a function $f\left(x,y,z\right)$ and sketch a region G in 3-space so that the triple integral in rectangular coordinates ${\displaystyle \underset{G}{\iiint }f\left(x,y,z\right)dV}$ matches the iterated integral in cylindrical coordinates given in part (a).