Double Integral Definition

Calculus 3 · Axiom Academy

LESSON Double Integrals: Definition & Properties From Riemann sums to signed volume — double integration over a rectangle, and Fubini's theorem. 1. From Riemann Sums to Double Integrals Take a function f(x,y) defined over a rectangle . To find the volume under the surface z = f(x,y) , we first partition R into small subrectangles. Split [a,b] into m pieces of width . Split [c,d] into n pieces of width . This makes mn subrectangles, each of area . In each subrectangle pick a sample point (x_ ij ^*, y_ ij ^*) and raise a box of height f(x_ ij ^*, y_ ij ^*) over the base . That one box has volume . Adding every box gives a Riemann sum : The double integral is the limit of these Riemann sums as the partition becomes infinitely fine — as and every . The area element dA can be written as or — the choice sets up the order of integration in Fubini's theorem below. Double integrals obey the same structural rules as single integrals, which lets us break a hard integral into easier pieces: The additivity property, shown above, is the workhorse: cut a region into non-overlapping pieces , integrate over each, and add — the foundation for integrating over shapes that are not rectangles. For a rectangle, the most powerful tool is Fubini's theorem : it evaluates a double integral as two ordinary integrals, one variable at a time. Inner integral: treat the outer variable as a constant. Evaluate it to get a function of the one remaining variable.

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