Calculus

Lesson 9 of 14

Multiple Integrals

Double and triple integrals, Fubini's theorem, and polar coordinates.

Volume under a surface

A double integral ∬Rf(x,y) dA\iint_R f(x,y)\,dA accumulates ff over a 2-D region. Geometrically, it is the volume under the surface z=f(x,y)z = f(x,y). Over a rectangle, Fubini's theorem lets you compute it as an iterated integral, in either order:

∫01 ⁣ ⁣∫02xy dy dx=∫01x[y22]02dx=∫012x dx=1\int_0^1\!\!\int_0^2 xy\,dy\,dx = \int_0^1 x\left[\frac{y^2}{2}\right]_0^2 dx = \int_0^1 2x\,dx = 1

In probability, integrating a joint density f(x,y)f(x,y) over a region gives the probability that (X,Y)(X,Y) lands in that region, and this is where multiple integrals earn their keep in quant interviews.