$ \\ \int_0^2\int_0^y\int_0^{\sqrt{4-y^2}}2xdxdzdy\\ \int_0^2\int_0^y4-y^2dzdy\\ \int_0^2(4-y^2)ydy\\ -\frac{1}{2}\left( \left.4y-\frac{1}{3}y^3\right|_0^2\right )\\ =-\frac{8}{3} $
I have worked over this problem several times and I cannot find the step where I went wrong.