Finding $\iiint_{D}x \, dV$ where $D=\left\{x^2+y^2\le1;x,y,z>0;x+y+z<1\right\}$

Question

I'm trying to solve the following triple integral using cylindrical coordinates:

$$\iiint_{D}x \, dV\,,$$ where $$D=\left\{x^2+y^2\le1;x,y,z>0;x+y+z<1\right\}$$

This is what I tried to do but I'm not sure how to continue or even if it is all correct. I would love to have you comments and assistance!

Answer 1

The cylinder doesn't matter at all in this problem because the tetrahedron formed by the plane $x+y+z=1$ and the coordinate planes is contained entirely within the cylinder. So the integral is given by

$$\int_0^1 \int_0^{1-x} \int_0^{1-x-y}x\:dz\:dy\:dx = \frac{1}{24}$$