Volume of a cylinder cut by a plane

Question

I've looked online but I can't seem to find a calculus proof for the volume of a cylinder cut by a plane. The question is:Write down a triple integral for the volume of the region of space in the 1st octant that is cut from the cylinder y2 + z2 = 1 by the planes y = x and x = 1. I convert to cylindrical coordinates but when I integrate X, which goes from x = 1 - x = y. I get stuck, I can solve the rest if I just get past this first step.

Answer 1

For given $x$ and $y$, the range in $z$ is from $0$ to $\sqrt{1-y^2}$. For given $x$, the range in $y$ is from $0$ to $x$. The range in $x$ is from $0$ to $1$.

Hence the volume of the solid is

$$\int_0^1\int_0^x\int_0^{\sqrt{1-y^2}}dz\ dy\ dx$$

Of course other orders of $x,y,z$ are possible.