Fubini's theorem

Statement

Let $R = [a, b] \times [c, d]$, and let $f \colon R \to \mathbb{R}$ be Riemann integrable on $R$. Suppose that $y \mapsto f(x, y)$ is integrable on $[c, d]$ for each $x \in [a, b]$, and that $x \mapsto f(x, y)$ is integrable on $[a, b]$ for each $y \in [c, d]$. Then $$\iint_R f \, dA = \int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy.$$ In particular, this holds for each continuous function $f$ on $R$.

Remarks

The left side is the double integral, and the other two are iterated integrals. The Lebesgue version: let $f$ be Lebesgue measurable on $\mathbb{R}^n \times \mathbb{R}^m$ with $\int |f| < \infty$. Then the double integral equals both iterated integrals, where the inner integrals exist for almost all values of the outer variable. Without integrability of $|f|$, the iterated integrals can differ. For non-negative functions, Tonelli's theorem removes the integrability hypothesis. Euler knew the case of continuous functions. Lebesgue proved a version for bounded measurable functions in 1904, and Fubini proved the version for integrable functions in 1907.

Explanations

L1 human

To integrate over a rectangle, you can slice it into lines, integrate along each line, and then integrate the results. Fubini's theorem says that both slicing directions give the same answer as the double integral, for well-behaved functions.