I don't really have too much trouble finding probabilities using joint probability density functions (PDFs) (of two variables) by drawing the area of support in the $xy$-plane, and then integrating under the PDF, $f(x,y)$, within the region of interest. So for example, If I were interested in finding $P(X+Y<1)$ for the PDF $f(x,y)=4xy$ on $0<x,y<1$, I would start by drawing the unit square, then drawing a line from the the point $(0,1)$ to $(1,0)$, and then I'd integrate under $4xy$ within the area represented by the triangle in the lower left of the unit square (e.g. $\int_0^1\int_0^{1-x}{4xy}\text{ }dydx$). No problem so far.

But what I've realized is that I'm only able really carry out these problems because I can appropriately and easily draw a graph to determine the area in which to integrate. For higher order dimensions, how does one normally go about finding probabilities without the use of a computer? I can't really draw out a graph for example in say, $4, 5, 6,$ or, say, $10$ dimensions. So, my question, really, is: how do statisticians carry out probability calculations with PDFs of high dimensions since you can't graph the area of support and area of interest as you could with 2 random variables?


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