Question

Evaluate x2 y2 dv, e where e is the region that lies inside the cylinder x2 y2 = 9 and between the planes z = 2 and z = 3.

Answer 1

Setting


`\begin{cases}x(r,\theta,\zeta)=r\cos\theta\\y(r,\theta,\zeta)=r\sin\theta\\z(r,\theta,\zeta)=\zeta\end{cases}`

you arrive at the Jacobian


`J=(\partial(x,y,z))/(\partial(r,\theta,\zeta))=\begin{vmatrix}x_r&x_\theta&x_\zeta\\y_r&y_\theta&y_\zeta\\z_r&z_\theta&z_\zeta\end{vmatrix}`

`J=\begin{vmatrix}\cos t&-r\sin t&0\\\sin t&r\cos t&0\\0&0&1\end{vmatrix}=r\end{vmatrix}`

Then the integral is


`\displaystyle\iiint_E(x^2+y^2)\,\mathrm dV=\int_(\zeta=2)^(\zeta=3)\int_(\theta=0)^(\theta=2\pi)\int_(r=0)^(r=3)r^2|J|\,\mathrm dr\,\mathrm d\theta\,\mathrm d\zeta`

`=\displaystyle\int_2^3\mathrm d\zeta\int_0^(2\pi)\mathrm d\theta\int_0^3r^3\,\mathrm dr`

`=(3-2)*(2\pi-0)*\frac14r^4\bigg|_(r=0)^(r=3)`

`=\frac{81\pi}2`