Question: Integrate the function $f(x,y,z)=y$ over the region R bounded by the plane $x+y+z=2$, the cylinder $x^2+z^2=1$, and the plane $y=0$.
My solution: since, $0<=y<=2-x-z$ and $x^2+z^2=1$
so let $x=r\cos{\theta}$ and $z=r \sin{\theta}$ and then I get the integration:
$$\int_{0}^{1} \int_{0}^{2 \pi} \int_{0}^{2-r\cos{\theta} - r\sin{\theta}} yr\,dy\,dr\,d\theta$$ I feel like I'm making this very complicated. Am i doing something wrong?
Your integral $$\int_{0}^{1} \int_{0}^{2 \pi} \int_{0}^{2-r\cos{\theta} - r\sin{\theta}} yr\,dy\,dr\,d\theta$$ should have been $$\int_{0}^{2 \pi}\int_{0}^{1} \int_{0}^{2-r\cos{\theta} - r\sin{\theta}} yr\,dy\,dr\,d\theta$$
Otherwise it is OK.