# Double integral setup -- volume of bounded cylinder

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.