[Math] Evaluate the triple integral $\iiint_E \sqrt{x^2+y^2}dV$

Question

Use cylindrical coordinates to evaluate the triple integral

$$\iiint_E \sqrt{x^2+y^2}dV, $$ where $E$ is the solid bounded by the circular paraboloid $z=16−4(x^2+y^2)$ and the $xy$-plane.
Please Help I am confused.

Answer 1

First draw a picture:

You then just need to set up the integral according to the picture as follows:

$$\int_0^{16} dz \: \int_0^{\sqrt{4-z/4}} dr \: r^2 \: \int_0^{2 \pi} d\theta$$

What is going on here? The volume element $dV = r\,dr\,d\theta\,dz$. The volume is rotationally symmetric as you can see, so there's no dependence on $\theta$. Note also that I choose to integrate disks parallel to the $xy$ plane through $z$; this involves solving for $r$ as a function of $z$. Note the extra factor of $r$ comes from your specification of the integral of $r \, dV$. Finally, we integrate over $z$ from $z=0$, i.e., the $xy$ plane, through to the top of the solid at $z=16$.

We then need to evaluate the integral. I'll reduce it to a single integral for you to evaluate:

$$\frac{2 \pi}{3} \int_0^{16}\: dz (4-z/4)^{3/2}$$

You should be able to do this one out.