Iterated Integrals and Fubini's Theorem

Core Idea

Fubini's theorem states that for continuous f on a rectangular region, ∬_R f(x, y) dA = ∫∫ f(x, y) dy dx (inner integral with respect to y, outer with respect to x). The two orders of integration give the same answer, providing flexibility in computation.

Explainer

From double integrals, you know that ∬_R f(x, y) dA represents a signed volume under the surface z = f(x, y) over a region R in the xy-plane. The abstract definition approximates this with Riemann sums over small rectangles, but actually computing that limit directly is unwieldy. Iterated integrals give you a concrete computational algorithm: integrate one variable at a time, treating the other as a constant during each step.

The intuition is a slicing argument. Fix a value x₀ and look at the "slice" of the region at that x-value: you get a one-dimensional cross-section, and ∫f(x₀, y) dy is the area under that slice (a signed area, counted by the function values). Now imagine sliding x₀ from left to right — you're sweeping out the entire region slice by slice. Integrating those slice areas ∫(∫f(x, y) dy) dx accumulates the total signed volume. Fubini's theorem makes this rigorous: for a continuous function on a rectangle [a, b] × [c, d], the double integral equals the iterated integral in either order:

∬_R f(x, y) dA = ∫_a^b (∫_c^d f(x, y) dy) dx = ∫_c^d (∫_a^b f(x, y) dx) dy.

The inner integral is evaluated first (treating the outer variable as a constant), then the result — a function of the remaining variable — is integrated by the outer integral.

The power of having two available orders becomes apparent when one order is computationally easier than the other. Consider integrating f(x, y) = e^y² over a triangular region where y ranges from x to 1 with x from 0 to 1. In the natural order (integrate y first), the inner integral is ∫_x¹ e^y² dy — a function with no closed form. Switching to integrate x first: for a fixed y, x runs from 0 to y, so the inner integral is ∫_0^y e^y² dx = y·e^y², which is easy. The outer integral ∫_0¹ y·e^y² dy evaluates to (e − 1)/2 by substitution. Switching the order of integration — by re-drawing the region and re-reading the bounds — is one of the most practically useful skills in multivariable calculus and appears constantly in probability, physics, and engineering applications.