Multivariate calculus from first principles
A single integral measured area under a curve. The double integral measures volume under a surface. Cover a region of the plane with tiny tiles, multiply each tile's area by the surface's height above it, add them up, then shrink the tiles. It's the Riemann-sum idea lifted into one more dimension.
You compute it by iterated integration: integrate over one variable, then the other. Fubini's theorem is what makes this practical, since for continuous functions you may integrate in either order and get the same answer.
Imagine measuring the total rainfall caught over a whole field. The rain falls unevenly, heavier near one corner, lighter at another, so you mentally chop the field into small squares, multiply each square's area by the local rainfall depth there, and add up every patch. Letting the patches shrink turns that sum into the double integral of the depth f(x, y) over the field.