Triple integral $\iiint x^2 \,dx\,dy\,dz.$

Question

Here's the question $$\iiint x^2 \,dx\,dy\,dz.$$ I am asked to evaluate this integral over the region $$D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2 \leq y^2+z^2 \leq 4\right \}.$$

I've shown that :

• $ 2 \leq x \leq -2 $

then:

$$\iiint x^2 \,dx\,dy\,dz= \int_{-2}^{2}dx \int\int x^2 dydz$$

From here,do i have to use the cylindrical coordinates?

Any support for this question would be appreciated.

Answer 1

I use cylindrical coordinates $y^2+z^2=r^2$ and $x=h$. Then $h$ varies between $-r$ and $r$. Then my integral is $$\iiint_D rdr d\theta h^2dh=\int_0^{2\pi}d\theta\int_0^2dr\ r\int_{-r}^rh^2dh$$