I need help setting up the bounds for this integral. The question is:
Let E be the solid between two cylinders : $x^2 + y^2 = 1$ and $x^2 + y^2 = 16$; and between two planes : $z = 0$ and $z = x+y+8$. Evaluate $\iiint_E x dV$.
I have it set up as:
$$\int_{0}^{\pi/2}\int_{1}^{4}\int_{0}^{x+y+8} x rdrd\theta$$
But for some reason I don't feel like that's right. Guidance is much appreciated! Thank you!
You need to run all the way around with $\theta$. You also have to convert to cylindrical coordinates where $x= r\cos(\theta)$ and $y=r\sin(\theta)$. But the rest is fine. So, using cylindrical coordinates you have the integral $$ \int_0^{2\pi}\int_1^4\int_0^{r\cos(\theta)+r\sin(\theta)+8} r\cos(\theta) r\; dz\; dr\; d\theta $$