Multivariate calculus from first principles
Add one more dimension and you have the triple integral: instead of tiling a 2-D region, you fill a 3-D solid with tiny boxes, weight each by the function's value there, and sum. The machinery is the same as before, Riemann sums followed by iterated integration, with Fubini still letting you choose the order.
Over a box [a,b]×[c,d]×[e,g] it's three nested single integrals: integrate over one variable holding the others fixed, then the next, then the last. Each step is ordinary Course-I integration.
Think of weighing a sponge cake whose density varies from place to place: airy near the top, denser and moister toward the middle. To get its total mass you'd dice it into tiny cubes, multiply each cube's little volume by the density right there, and add every crumb. Shrinking the cubes turns that sum into the triple integral of the density f(x, y, z) over the cake.