Problem (more here and the problem XIV.6:6 here on page 976)
Integrate $\int_A z dx dy dz$ in cylinder-coordinates when $A=\{(x,y,z)\in\mathbb R^3 | x^2+y^2 \leq z \leq \sqrt{2-x^2-y^2}\}.$
So $r^2=x^2+y^2$ and
$\int_0^{2\pi} \int_a^b \int_{r^2}^{\sqrt{2-r^2}} rz dz dr d\rho$ but what $a$ and $b$ are? I know for the max -case $0\leq z \leq \sqrt{2}$ and $0\leq r \leq 2^{0.25}$ if $z,r\in\mathbb R$. $\int_0^{2\pi} \int_0^{2^{0.25}} \int_{r^2}^{\sqrt{2-r^2}} rz dz dr d\rho,$ just this easy? With this specification, I got $\pi \left( \sqrt{2} - 0.5-\frac{1}{6} 2^{1.5} \right)$ as a solution. But if wrongly specified, it is not right. Perhaps, I need to find an expression for the $r$ in terms of the angle $\rho$ -- somehow from $x=r \cos(\rho)$, $y=r\sin(\rho)$ and $z=z$? I may be misunderstanding here the exercise, perhaps I am just over-engineering...