Using whatever method you like, evaluate the integral $I =\int \int \int_ExdV$ where $E$ is the region between the cylinders $x^2+y^2=1$ and $x^2+y^2=4$, below the plane $z=y+2$ and above the $xy$-plane.
My attemp at setting this up. I make the substitution $dV=dzdydx$ I believe that this is the easiest way. This way I don't have to change the $x$ in the integrand. I get this integral.
$$\int_{?}^? \int_{?}^? \int_{?}^? xdzdydx$$
I still have to find the limits of integration. to find $y$ I do some calculations. $$x^2+y^2=4, x^2+y^2= 1$$ $$x^2=1-y^2, (1-y^2) + y^2=4$$ Solve for $y$ $$(y^2-y^2)+ 1 = 4$$ $$1=4?$$
This does not add up. I am having trouble finding the limits of integration. If anyone can help me that would be great. Thanks
Also can I go this route?
$$dV=dxdydz$$
$$\int_{0}^2 \int_{-\sqrt{1-x^2}}^\sqrt{1-x^2} \int_{-\sqrt{1-y^2}}^\sqrt{1-y^2} xdxdydz$$
This is another one of my thoughts. Maybe $z$ is a max at 2 because at the $xy$-plane $z=0$ if $y=0$, so just plug $y=0$ into the equation. $z=(0)+2$, then $z=2$ Am I on the right track with this one?