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Fubini's Theorem — Definition, Formula & Examples

Fubini's Theorem states that if a function is continuous (or absolutely integrable) over a rectangular region, you can evaluate a double integral by integrating one variable at a time in any order.

Let f(x,y)f(x,y) be integrable on the rectangle R=[a,b]×[c,d]R = [a,b] \times [c,d]. If ∫R∣f(x,y)∣ dA<∞\int_R |f(x,y)|\,dA < \infty, then the double integral equals either iterated integral: ∬Rf(x,y) dA=∫ab ⁣∫cdf(x,y) dy dx=∫cd ⁣∫abf(x,y) dx dy\iint_R f(x,y)\,dA = \int_a^b \!\int_c^d f(x,y)\,dy\,dx = \int_c^d \!\int_a^b f(x,y)\,dx\,dy.

Key Formula

∬Rf(x,y) dA=∫ab(∫cdf(x,y) dy)dx=∫cd(∫abf(x,y) dx)dy\iint_R f(x,y)\,dA = \int_a^b \left(\int_c^d f(x,y)\,dy\right)dx = \int_c^d \left(\int_a^b f(x,y)\,dx\right)dy
Where:
  • RR = Rectangular region $[a,b] \times [c,d]$
  • f(x,y)f(x,y) = An integrable function on $R$
  • a,ba, b = Limits of integration for $x$
  • c,dc, d = Limits of integration for $y$

How It Works

To apply Fubini's Theorem, first confirm that ff is continuous on the region or that ∫R∣f∣ dA\int_R |f|\,dA is finite. Then choose an order of integration—either dx dydx\,dy or dy dxdy\,dx—and evaluate the inner integral while treating the other variable as a constant. Finally, evaluate the outer integral. When the region is not rectangular, you can still swap the order, but you must carefully rewrite the limits of integration to match the new order.

Worked Example

Problem: Evaluate ∬R(x+2y) dA\iint_R (x + 2y)\,dA where R=[0,1]×[0,2]R = [0,1] \times [0,2].
Step 1: Set up the iterated integral by integrating with respect to yy first.
∫01∫02(x+2y) dy dx\int_0^1 \int_0^2 (x + 2y)\,dy\,dx
Step 2: Evaluate the inner integral, treating xx as a constant.
∫02(x+2y) dy=[xy+y2]02=2x+4\int_0^2 (x + 2y)\,dy = \left[xy + y^2\right]_0^2 = 2x + 4
Step 3: Evaluate the outer integral.
∫01(2x+4) dx=[x2+4x]01=1+4=5\int_0^1 (2x + 4)\,dx = \left[x^2 + 4x\right]_0^1 = 1 + 4 = 5
Answer: The value of the double integral is 55.

Why It Matters

Fubini's Theorem is the workhorse of multivariable integration—virtually every double or triple integral you compute in a calculus course relies on it. In physics and engineering, it underlies calculations of mass, center of mass, and probability over multi-dimensional regions. Choosing a clever order of integration often turns an otherwise intractable integral into a routine computation.

Common Mistakes

Mistake: Swapping the order of integration without checking integrability or adjusting the limits.
Correction: Fubini's Theorem requires ff to be absolutely integrable on the region. When the region is non-rectangular, changing the order demands rewriting the integration bounds to describe the same region in the new variable order.