I have to solve the integral $$\iiint_D \frac{y}{1+\sqrt{z}}\,dx\,dy\,dz$$ over the region $$D=\{(x,y,z)\in\mathbb{R^3}: x^2+y^2\le z\le 1\}$$.
I parameterized the region with cylindrical coordinates $$\cases{x=r\cos t\\y=r\sin t\\z=z}$$ and the boundaries I put are $0\le{t}< 2\pi, 0\le r\le 1, r^2\le z\le 1$. So the integral is $$\int_0^{2\pi} \int_0^1 \int_{r^2}^1 \frac{r\sin t}{1+\sqrt z}\,dz\,dr\,dt=\int_0^1 r \int_0^{2\pi}\sin t\int_{r^2}^1 \frac{1}{1+\sqrt z}\,dz\,dt\,dr$$
Can I conclude that the total triple integral is 0, given that the last integral doesn't depend on t which leads to the integral of $\sin t$ being 0?
You can conclude that the integral is $0$ from the original integral itself, using the fact that $y$ is an odd function.