Complex task with cylinder and its part Let r be a positive constant. Consider the cylinder $x^2+y^2 \le r^2$, and let C be the part of the cylinder that satisfies $0 \le z \le y$. 
1) Consider the cross section of C by the plane x=t ($-r \le t \le r$), and express its area in terms of r, t.
2) Calculate the volume of C, and express it in terms of r.
3) Let a be the length of the arc along the base circle of C from the point (r, 0, 0) to the point ($r \cos φ, r \sin φ, 0$) ($0 \le φ \le \pi$). Let b be the length of the line segment from the point ($r \cos φ, r \sin φ, 0$) to the point ($r \cos φ, r \sin φ, r \sin φ$). Express a and b in terms of r, φ.
4) Calculate the area of the side of C with $x^2+y^2=r^2$, and express it in terms of r.

Good day! 
I hadn't met this type of problem before. Honestly, I don't understand what these variables mean. Do they mean coordinates? I'm confused. I want you to explain how to solve this type of problems (I'll be very happy, if you add a drawing of this cylinder with C part and its cross section). Please, help me. 
Thanks in advance!
 A: Yes, the variables are coordinates. Let's say that our coordinate system is such as $xy$ are in the horizontal plane, and $z$ is vertical direction. Then you have a cylinder with the axis along the $z$ direction, with radius $r$. Think of it like a glass sitting on a table. Now look at $C$. The first inequality says $z\ge 0$, so we are interested in only the upper part of the cylinder. The second inequality says $z\le y$. That means that $z=y$ is an upper bound. This is an equation of a plane at $45^\circ$ with the horizontal. Notice also that $y\ge 0$. So you take your "glass", choose only half of it $(y\ge0)$, then cut it at $45^\circ$.
For part 1, you also cut it with a constant plane, parallel to the $yz$ plane, at a distance $t$ from the origin. Notice that if $t>r$ or $t<-r$ you don't have an intersection. So the cross section is a right angle isosceles triangle. The length of the side is $\sqrt{r^2-t^2}$
For part 2, write the volume as a sum of many such triangles, with a small width $\Delta t$. You can transform it to an integral.
Part 3 and 4 ask you to do similar things, but in polar coordinates
A: Attached is my solution with a sketch:

