Fubini’s Theorem

Summary

Fubini’s theorem justifies evaluating a double (or triple) integral as an iterated integral, integrating one variable at a time. On a rectangle, if $f$ is continuous, the order of integration may be swapped freely.

Prerequisites

Integrals, Riemann Sum, Double Integrals

Theorem

Continuous functions on a rectangle

Let $f$ be continuous on $R = [a, b] \times [c, d]$ . Then $f$ is integrable on $R$ and

\iint_{R} f (x, y) d A = \int_{a}^{b} \int_{c}^{d} f (x, y) d y d x = \int_{c}^{d} \int_{a}^{b} f (x, y) d x d y .

Type I and Type II regions

For a Type I region $D = {(x, y) : a \leq x \leq b, g_{1} (x) \leq y \leq g_{2} (x)}$ with $f$ continuous on $D$ ,

\iint_{D} f d A = \int_{a}^{b} \int_{g_{1} (x)}^{g_{2} (x)} f (x, y) d y d x .

For a Type II region $D = {(x, y) : c \leq y \leq d, h_{1} (y) \leq x \leq h_{2} (y)}$ ,

\iint_{D} f d A = \int_{c}^{d} \int_{h_{1} (y)}^{h_{2} (y)} f (x, y) d x d y .

When both descriptions apply, either order is valid; choose the easier limits.

Conditions / Assumptions

Continuity on a compact region is the elementary sufficient condition.
If $f$ is only integrable (or improper), more care is needed; absolute integrability allows reordering in advanced settings.

Worked Example

On $[0, 1] \times [0, 1]$ ,

\int_{0}^{1} \int_{0}^{1} (x + 2 y) d y d x = \int_{0}^{1} [x y + y^{2}]_{0}^{1} d x = \int_{0}^{1} (x + 1) d x = \frac{3}{2},

and the reverse order yields the same value.

Common Mistakes

Changing order without redrawing the region and updating limits.
Applying Fubini to a non-integrable singularity without checking absolute convergence.

Connections

Double Integrals, Double Integrals over General Regions, Triple Integrals

References

Fubini’s theorem and iterated integrals appear in OpenStax Calculus Volume 3.^[1]

OpenStax, Calculus Volume 3, Section 5.1–5.2, https://openstax.org/details/books/calculus-volume-3 ↩︎