I have the following problem:
Let $0 ≤ R ≤ 1.$ Compute the volume of the following 3-dimensional shape: the intersection of the cylinder given by $x^2 + y^2 ≤ R^2$ and the ball of radius 1 centered at the origin.
We also got this clarification: Think of R as any fixed value between 0 and 1. R is not the z coordinate. For every R there is a 3-D shape and a volume to compute. The answer is a function of R. For example, for R=1/2 you want the volume of the intersection of a cylinder of radius R and a ball of radius 1.
How can I set up this problem? I have no clue on how to approach it.
Let $R$ be fixed at some value $0 < R \le 1$. Then the upper limit on $z$ would be $z= \sqrt{1-(x^2 + y^2)}$. In cylindrical coordinates this would be $z = \sqrt{1-r^2}$. The limits on $r$ would then be $0 \le r \le R$ and $0 \le \theta \le 2 \pi$. If you use $0$ for the lower limit on $z$ and then multiply the integral by $2$, you will have the answer. Can you write the integral now?