Fubini's Theorem for Double Integrals

Definition

Fubini’s Theorem states that if $f(x, y)$ is continuous on a rectangular region $R: a \leq x \leq b, c \leq y \leq d$, the double integral can be evaluated as iterated single integrals in either order: $\iint_R f(x, y) dA = \int_c^d \int_a^b f(x, y) dx dy = \int_a^b \int_c^d f(x, y) dy dx$

• How to read: “The double integral of f of x y over the region R equals the iterated integral from c to d and a to b of f with respect to x then y; which is also equal to the iterated integral from a to b and c to d of f with respect to y then x.”
• Meaning: The total accumulated value over a rectangular region can be computed by slicing the region one variable at a time. For continuous $f$, either integration order gives the same answer.

Why It Matters

Fubini’s Theorem is the essential ‘shortcut’ for multivariable calculus; by allowing us to swap the order of integration, it transforms complex, seemingly impossible 2D accumulation problems into straightforward 1D calculations, saving countless hours of algebraic frustration in engineering and physics.

Core Concepts

• Fubini’s Theorem (iterated form) $\iint_R f(x, y) \, dA = \int_c^d \int_a^b f(x, y) \, dx \, dy = \int_a^b \int_c^d f(x, y) \, dy \, dx$
  • How to read: “The double integral over the region R equals the integral with respect to x from a to b on the inside, and the integral with respect to y from c to d on the outside, or vice versa.”
  • Meaning / when to use: Turn a 2D accumulation problem into two nested 1D integrals. Pick whichever order makes the algebra simpler; both are valid when $f$ is continuous on the rectangle $R$.

• Iterated Integration: Reducing a multivariable problem into a sequence of single-variable integrations.
• Order Independence: For continuous functions, the sequence of integration (dx then dy, or dy then dx) yields the same result.
• Continuity Requirement: The theorem relies on the function being continuous over the region of integration.

Connected Concepts

• Mean Value Theorem for Integrals
• Comparison Theorem for Improper Integrals
• Area by Double Integration and Average Value by Double Integration