I am confused about how to find the bounds of integration for the following triple integral. I seem to have trouble figuring out how to parameterize these curves (like the paraboloid, below).
Also, based on my parameterization, the function that I am integrating over (mass density function, $x^2+y^2$, how does that change once I parameterize it?
$$\int \int \int (x^2+y^2)dxdydz$$ where $$ \\~\\y^2+z^2 \leq x, \\~\\ 0 \leq x \leq h$$
EDIT: Would the $y$ bounds go from $0$ to $r cos (\theta)$, and the $z$ bounds go from $0$ to $r sin (\theta)$? I am pretty sure the $x$ bounds go from $0$ to $h$. But from here, do I plug in $x=h$ into $x^2$ and $y=rcos(\theta)$ for $y^2$?
With $y=r\cos\theta$, $z=r\sin\theta$, Jacobian $r$, $$ \int_0^h dx \int_{y^2+z^2\le x} dy dz (x^2+y^2) $$ $$ = \int_0^h dx \int_{0}^{\sqrt{x}} r dr \int_0^{2\pi} d\theta (x^2+r^2\cos^2\theta) $$ $$ = \int_0^h dx \int_{0}^{\sqrt{x}} r dr (2\pi x^2+\pi r^2) $$ $$ = \int_0^h dx (\frac{\pi}{4}x^2+\pi x^3) $$ $$ = \frac{\pi}{12}h^3(1+3h). $$