Originally Posted by
HallsofIvy I think you mean "y^2+ z^2= 1". I am not clear what the two "0" s mean. And I think you mean the planes "y= x" and "y= z".
One "usually" sees the cylinder given as x^2+ y^2= 1. That is a cylinder with axis the z-axis and whose cross section at the xy-plane is the circle centered at (0,0,0) and radius 1. That region can be parameterized by $\displaystyle x=r cos(\theta)$, $\displaystyle y= r sin(\theta)$. z= t. So it should be clear that, here, the rolls of "x" and "z" have been swapped. This is a cylinder with axis the x-axis and whose cross section at the yz-plane is the circle centered at (0, 0, 0) and radius 1.
Actually, what you are concerned with here is irrelevant! You can parameterize a circle with $\displaystyle x= r cos(theta)$ and $\displaystyle y= r sin(\theta)$ or $\displaystyle x= r sin(\theta)$ and $\displaystyle y= r cos(\theta)$. The only difference is that in the first $\displaystyle \theta= 0$ is the point (1, 0) while, in the second, it is (0, 1). But as $\displaystyle \theta$ goes from 0 to $\displaystyle 2\pi$, they both give the same circle.
The crucial point in this problem is that "x" and "z" have been reversed.
(You could get exactly the same answer by writing the problem as "find the volume bounded by cylinder x^2 + y^2 =1 , plane y = z and y= x plane in first octant...", by reversing "x" and "z" in the problem.)