Charged Pole

Begin with ${Q}_{enclosed} = \iiint { \rho dV }$ , which in this case is ${Q}_{enclosed}=\int _{ 0 }^{ h }{ \int _{ 0 }^{ 2\pi }{ \int _{ 0 }^{ R }{ 100{ e }^{ -z }{ \left( { x }^{ 2 }+{ y }^{ 2 } \right) }^{ -1/4 }rdrd\theta dz } } } .$

We switch to cylindrical coordinates $\left( { x }^{ 2 }+{ y }^{ 2 } \right) ={r}^{2}$ ; hence, ${Q}_{enclosed} = \int _{ 0 }^{ h }{ \int _{ 0 }^{ 2\pi }{ \int _{ 0 }^{ R }{ 100{ e }^{ -z }{ \left(r \right) }^{ 1/2 }drd\theta dz } } } .$

The result is then ${Q}_{enclosed}=\frac{400 \pi}{3} \left(1-{e}^{-h} \right) {R}^{3/2}.$