Integrals Over Flat Regions
A double integral adds up the values of a function across a two-dimensional region, one thin rectangle at a time.
Slicing two ways
Just as a single integral sums thin rectangles under a curve, a double integral sums thin boxes under a surface over a region R. Each box has base area dA and height f, so the integral measures signed volume when f is a height. If f is instead a density, the same integral gives total mass over the region.
\iint_R f(x,y)\,dAIterated integrals
In practice you compute a double integral as an iterated integral, doing the inner integral first while treating the outer variable as constant. For the rectangle example integrating xy over 0 to 2 in y and 0 to 3 in x, the inner integral gives 2x, and integrating that from 0 to 3 gives 9. Two nested single integrals replace one intimidating double symbol.
\int_0^3 \int_0^2 xy\,dy\,dx = 9Swapping the order
The area element dA can be read as dx dy or dy dx, and the two orders give the same answer when the region allows it. Switching order is a favorite trick because one order may have an antiderivative the other lacks. For non-rectangular regions, describe the boundary as functions and let the inner limits depend on the outer variable.
dA = dx\,dy = dy\,dxFlashcard
Signed volume under the surface over region R
Check your understanding
Evaluate the iterated integral of x, integrated over y from 0 to 3 and then over x from 0 to 2.