Evaluate $$\iiint_{D}(x^2+y^2)^2\,dx\,dy\,dz\,,$$ where $$D=\{(x,y):x^2+y^2\leq 1, 0\leq z\leq 1,0\leq y\}$$ I used GeoGebra to represent my domain. It is that right section of the cylinder, enclosed by the planes $z=0$, $z=1$. As I have a cylinder, is it a good ideea to use polar coordinates or should I use a projection of the domain on xOy? In fact, when I know which to use? Could you solve this as an example, I don`t have much. Thank you, it would be great.
In this particular example, it seems optimal to use polar coordinates, since you can represent $x^2+y^2$ in a very simple and covinient way. Let $(x,y,z)$ be a point in $\mathbb{R}^3$. You first set $x=r\cos\theta$ and $y=r\sin\theta$ where $\theta$ is the angle of the vector and the "positive" part $x$-axis and $r$ is the length of the vector (in genereal $r>0$ and $\theta \in [0, 2\pi]$). You basically use polar coordinates for the plane and leave the third coordinate untouched. With this setting and the given $D$, $x^2 + y^2=r^2$, and thus $0\leq r\leq 1$ and since $y\geq 0$, $\theta \in [0, \pi]$. Also the Jacobian of this transformation, which we need in order to change variables in the integral, is $r$.
Alos we can use Fubini since it is a continuous function on a compact set and thus it's integrable.
\begin{align} &\iiint_{D}(x^2+y^2)^2dxdydz = \int_0^1 \int_0^{\pi} \int_0^1 r^4 \cdot r \text{ d}z \text{ d}\theta \text{ d}r = \int_0^1 \int_0^{\pi}r^5 \text{ d}\theta \text{ d}r \\&= \int_0^1 \int_0^{\pi} r^5 \text{ d}\theta \text{ d}r = \pi \int_0^1 r^5 \text{ d}r = \dots \end{align}