$$\iiint_E z\sqrt{x^2+y^2}\,dx\,dy\,dz\,,$$ where $E$ is bounded by $y=\sqrt{4x-x^2}$ and $y=0, z=0,z =4$.
How can I calculate this? I found $E$ just like a half cylinder. Then I found $$0\leqq x\leqq4, 0\leqq z\leqq4, 0\leqq y\leqq\sqrt{4x-x^2}$$
But I don't know whether it is correct. What is the value of it?
Note
$$\iiint_E z\sqrt{x^2+y^2}\,dx\,dy\,dz = \int_0^4 zdz\int_0^{\pi/2} \int_0^{4\cos\theta} r^2drd\theta =\frac{1024}9 $$