Finding expected value for random variable $X$ given a joint probability density function $f(x,y)$ I've been given $f(x,y) = 6y$ with boundaries $0 \leq y \leq x \leq 1$.
How do I find the expected value of $x$?

$0\leq y \leq x\leq 1$ is the interval $\{(x,y)\in[0,1]\times [0, x]\}$ or $\{(x,y)\in [y,1]\times [0,1]\}$
Hence
$$\begin{align}
\mathsf E[X] & = \iint_{0\leq y\leq x \leq 1} x f(x,y)\operatorname d y\operatorname d x
\\[2ex] & = \int_0^1 x \int_0^x 6y \operatorname d y\operatorname d x
\\[2ex] & = \Box
\end{align}$$
