Find triple integral $\iiint_V \sqrt{x^2+y^2}sin(z^2) dxdydz$

Question

Find triple integral $\iiint_V \sqrt{x^2+y^2}sin(z^2) dxdydz$ when V: $\sqrt{x^2+y^2}\le z \le 3$.

First, I try to change variables $x=rcos\phi,y=rsin\phi,z=z$ but I can't find $\int sin(z^2)dz$. I try to let $z^2=u$ but it doesn't help. I think I have to find some ways to change variables which J ( Jacobi ) include z so I can caculus $\int zsin(z^2)dz$.

But no matter how many times I try, I still can't find the answer. So, can you guys help me with this exercise or give me any ideas? Thank you so much for your help.

Answer 1

In cylindrical coordinates, you have\begin{align}\int_0^{2\pi}\int_0^3\int_0^zr^2\sin(z^2)\,\mathrm dr\,\mathrm dz\,\mathrm d\theta&=2\pi\int_0^3\frac13z^3\sin(z^2)\,\mathrm dz\\&=\frac{1}{3} \pi (\sin (9)-9 \cos (9)).\end{align}Note that you can compute$$\int z^3\sin(z^2)\,\mathrm dz$$doing $z^2=x$ and $2z\,\mathrm dz=\mathrm dx$.