8.2 due on 11/30

Difficult
In the ellipse example, I'm not sure how |C| = (pi) ab. Where did the pi come from? I'm assuming it has to do with some equation for an ellipse, but I'm not sure. I'm also a little confused how P = (arctan(a/b)) / pi. I know the book says it's because Theta is uniformly distributed, but I still don't know how that worked out. Do we take an integral? Or perhaps it's something conceptual?
Reflective
In the last few sections, I realized how out of practice I am with derivatives and integration. In this section, the introduction of the Jacobian makes me feel even more out of practice. The theorem for the joint density of change of variables looks a lot like something from one of the Math 30-series classes (I'm not even sure which one!). So, even though this section wasn't incredibly difficult concept-wise, I feel it'll still be challenging because I'm so rusty with these kinds of calculations.