Consider the following joint PDF:
$$f_{X,Y}(x,y)=\begin{cases} 1 & \text{if } 0<x<1, -x<y<x \\ 0 & \text{otherwise}\end{cases}$$
I am able to visualize the figure in 3 dimensions as a triangle at height 1 spanning $y=-1$ to $1$ and $x=0$ to $1$. I am trying to study the properties of this PDF, such as conditional expectation, marginal PDF, etc., and I am having a really tough time properly setting up the integrals. Specifically, I want to know $E[Y|X]$ and the marginal PDFs $f_X(x)$, $f_y(y)$, so that I can study the unconditional expectations and covariance. I am frustrated because I cannot visualize what is going on when I compute these objects. Any hints would be very helpful.