Cylinder defined on 3d coordinate plane This is the first time, I have seen a problem like this:

I feel as though if i knew where to start i would be able to do this problem easily. In other words, question 1-4 make sence to me and i know what they are asking for, but i just can't visualize the cylinder. I'm not asking for a picture(although that would be nice), but clarification on what the question is telling me would be helpful. Thanks!

I don't have acesss to a plotting software or a scanner right now so I can't provide a precise plot, but you have the following ingriedents:


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*The equation $x^2 + y^2 = r^2$ is the equation of an infinite cylinder of radius $r$ whose symmetry axis is the $z$-axis. The inequality $0 \leq x^2 + y^2 \leq r^2$ throws in all the points inside the cylinder and so defines a solid cylinder.

*The equation $z = y$ is the equation of a plane in $\mathbb{R}^3$ that passes through the origin. The inequality $0 \leq z \leq y$ describes the region that lies below the plane and above the $xy$-plane.


The solid $C$ you are interested in lies below the plane $z = y$, inside the cylinder $x^2 + y^2 = r^2$ and above the $xy$ plane. The following image, taken from math.tutorvista.com shows a similar situation:

In the picture, $r = 1$ but the plane is $y + z = 2$ and not $z = y$ like in your scenario.
The first part of your question asks you to describe the cross section of $C$ by a plane $x = t$ which is the plane parallel to the $yz$ plane that passes through $(t,0,0)$. In the cross section, $x$ is constant and so the cross section is described by
$$ C_{t} = \left \{ (t, y, z) \, | \, y^2 \leq r^2 - t^2, \, 0 \leq z \leq y \right \}. $$
If you draw the inequalities defining $C_{t}$ in the $yz$ plane, you'll see that $C_{t}$ looks like a solid triangle. This is not surprising as the intersection of the solid cylinder with the plane $x = t$ is a strip and then choosing the points that lie above the $xy$ plane and below the plane $z = y$ will result in a solid triangle (try to imagine and draw this in 3d). This will allow you to do the first part of the question. For the second part of the question, you need to integrate the areas of the cross sections $C_t$ in the range $-r \leq t \leq r$ in order to get the volume of the solid $C$. 
