TITLE: Uniform distribution on unit disk
QUESTION [6 upvotes]: Let $(X, Y)$ be a random point chosen according to the uniform distribution in the disk of radius 1 centered at the origin. Compute the densities of $X$ and of $Y$.

I know that the joint density of $X$ and $Y$ is $\frac{1}{\pi}$ since when we integrate $\frac{1}{\pi}$ over the unit circle, we get $1$. 
So if I wanted to find the density of $X$, I was thinking of finding the cumulative distribution of $X$ and the differentiate it to get its density. In order to get its cumulative distribution function, I was going to use the fact that $P(X<x)=P(X<x, -\infty < Y < \infty)$, but this integral doesn't seem nice to work with. Am I on the right track or is there a better way?

REPLY [3 votes]: $P(X<x)=\int_{-1}^x\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\frac{1}{\pi}\text{d}y\text{ d}x=\int_{-1}^x\frac{1}{\pi}2\sqrt{1-x^2}\text{d}x$, so differentiating you just get back the inside $f_X(x)=\frac{1}{\pi}2\sqrt{1-x^2}$.