How to approach the calculations of probabilities in high dimensions?

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 $$0, I would start by drawing the unit square, then drawing a line from the the point $$(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?