I just want to check that I am understanding and setting up a double integral problem correctly.
The problem is: Find the volume of the region bounded by the cylinder $y^2 + z^2 = 4$ and the planes $x = 2y$, $x = 0$, $z = 0$ in the first octant.
So I set $z = \sqrt{4 - y^2}$ and then set the following boundaries: $0\leq x \leq 2y$ and $0 \leq y \leq 2$. Then the integral I put together looks like: $$\int_{0}^{2} \int_{0}^{2y} \sqrt{4-y^2}dxdy$$
Which is not a terribly difficult integral to solve, afterwards. But I wanted to make sure that this would be the correct setup. Thanks !
I have checked your work and your integras $$V=\int_{0}^{2} \int_{0}^{2y} \sqrt{4-y^2}dxdy$$ is set up correctly.
As you have mentioned, the integral is rather an easy one to evaluate.