How do I do this triple integration?

Question

I have to find the volume of the region in the first octant $(x\ge0,y\ge0,z\ge0$ bounded by the cylinder $x^2+y^2=4$ and the planes $z=2$ and $y+z=4$. I’m trying to evaluate the following integral to get the volume — $\int_0^2$ $\int_0^{4-z}$ $\int_0^{\sqrt{4-y^2}}$ $dx dy dz$

If I’m correct about the above, its evaluation is quite complicated... Is there an easier way to do this?

Answer 1

$$\int\limits_{x=0}^2 \int\limits_{y=0}^{\sqrt{4-x^2}} \int\limits_{z=2}^{4-y}\ 1\ dx\ dy\ dz = 2 \pi -\frac{8}{3}$$