Calculate the integral $\iiint_G(x^2 + z^2)\,\mathrm{d} x\,\mathrm{d} y\,\mathrm{d} z$

Question

Calculate the integral $$\iiint_G(x^2 + z^2)\,\mathrm{d} x\,\mathrm{d} y\,\mathrm{d} z$$ where $G$ is bounded by $2y=x^2+z^2$, $y=2$.

Please, give me some hints, how I must calculate this integral?

Answer 1

First, $y= (x^2+ z^2)/2$ is a paraboloid with vertex at (0, 0, 0) and axis of symmetry the y-axis. y= 2 is simply the plane parallel to the xz-plane "capping that paraboloid. I think I would be inclined, because of the symmetry, to use "cylindrical coordinates" but swapping "y" and "z": $x= r cos(\theta)$, $y= y$, $z= rsin(\theta)$. Then the boundary becomes $y= (x^2+ z^2)/2= r^2/2$, and y= 2. $y= r^2/2= 2$ when $r^2= 4$ or r= 2. We must take r from 0 to 2, $\theta$ from 0 to $2\pi$, and y from the paraboloid to the "cap", $r^2/2$ to 2. The integrand, $x^2+ z^2$ is just $r^2$, and $dxdydz= r dydrd\theta$.

The integral is $\int_0^{2\pi}\int_0^2\int_{r^2/2}^2 r^3 dydrd\theta$.

Answer 2

I'd suggest using cylindrical polar coordinates aligned along the $y$-axis, i.e. transform the integral in terms of the coordinates $(\rho, \theta, y)$, where $x^2+z^2=\rho^2$, $x=\rho \cos \theta$ and $z=\rho \sin \theta$.