9.3 Fubini's theorem and iterated integrals

Fubini's Theorem and Iterated Integrals

Definition and Concept of Fubini's Theorem

Fubini's theorem gives you a practical way to evaluate double integrals: instead of integrating over a two-dimensional region all at once, you can perform two successive single-variable integrations. This is the core technique you'll rely on for nearly every double integral computation in this course.

The formal statement: if $f(x,y)$ is continuous on a closed, bounded rectangular region $R = [a,b] \times [c,d]$, then

$\iint_R f(x,y)\, dA = \int_a^b \int_c^d f(x,y)\, dy\, dx = \int_c^d \int_a^b f(x,y)\, dx\, dy$

Both iterated integrals give the same result, and both equal the double integral. This means you get to choose which variable to integrate first.

Iterated Integrals and Interchange of Integration Order

An iterated integral is just a nested pair of single-variable integrals. In $\int_a^b \int_c^d f(x,y)\, dy\, dx$, you integrate with respect to $y$ first (treating $x$ as a constant), then integrate the result with respect to $x$.

Fubini's theorem guarantees that swapping the order of integration doesn't change the answer, as long as $f$ is continuous on the rectangle. This flexibility is genuinely useful: sometimes integrating $dy\, dx$ leads to a nasty antiderivative, while $dx\, dy$ works out cleanly (or vice versa). When you're stuck on an iterated integral, try switching the order before reaching for more advanced techniques.

Representing Double Integrals as Repeated Single Integrals

Here's the step-by-step process for evaluating a double integral using Fubini's theorem:

1. Set up the iterated integral. Choose an order of integration and write the appropriate limits. For a rectangle $R = [a,b] \times [c,d]$, the limits are constants.
2. Evaluate the inner integral. Integrate with respect to the inner variable, treating the outer variable as a constant. You'll get a function of the outer variable only.
3. Evaluate the outer integral. Integrate the result from step 2 with respect to the outer variable. This gives you a number: the value of the double integral.

Example: Evaluate $\iint_R x^2 y\, dA$ where $R = [0,2] \times [0,3]$.

Choosing $dy\, dx$ order:

$\int_0^2 \int_0^3 x^2 y\, dy\, dx = \int_0^2 x^2 \left[\frac{y^2}{2}\right]_0^3 dx = \int_0^2 x^2 \cdot \frac{9}{2}\, dx = \frac{9}{2}\left[\frac{x^3}{3}\right]_0^2 = \frac{9}{2} \cdot \frac{8}{3} = 12$

Conditions for Fubini's Theorem

Continuity Requirement

Fubini's theorem requires $f(x,y)$ to be continuous on the region of integration. Continuity means the function has no jumps, holes, or undefined points within the region.

If continuity fails, the two iterated integrals can actually give different values, which means neither one reliably represents the double integral. A classic example: define

$f(x,y) = \frac{x^2 - y^2}{(x^2 + y^2)^2}$

for $(x,y) \neq (0,0)$. The two iterated integrals over a region containing the origin can yield different results, showing that Fubini's theorem genuinely breaks down without continuity.

In practice, check for points where the function blows up or is undefined. If those points lie inside or on the boundary of your region, you can't apply Fubini's theorem directly.

Bounded Region Requirement

The region of integration must also be bounded, meaning it fits inside some finite rectangle in the $xy$-plane. For this unit, you're working with rectangular regions described by $a \leq x \leq b$ and $c \leq y \leq d$, where $a, b, c, d$ are all finite numbers.

If the region extends to infinity (like $R = \{(x,y) \mid x > 0,\, y > 0\}$), the integral becomes an improper double integral. These require separate convergence analysis and can't be handled by Fubini's theorem alone.

Applications

Applications in Physics

Moments of inertia for flat objects use the integral $I = \iint_R \rho(x,y)\, r^2\, dA$, where $\rho(x,y)$ is the mass density and $r$ is the distance from the axis of rotation. Fubini's theorem turns this into two manageable single integrals.

Center of mass calculations follow a similar pattern. For a lamina with density $\rho(x,y)$ over region $R$:

• Total mass: $M = \iint_R \rho(x,y)\, dA$
• Coordinates: $\bar{x} = \frac{1}{M}\iint_R x\,\rho(x,y)\, dA$, and $\bar{y} = \frac{1}{M}\iint_R y\,\rho(x,y)\, dA$

Each of these double integrals gets evaluated as an iterated integral via Fubini's theorem.

Applications in Engineering

Volume of a solid beneath a surface $z = f(x,y)$ and above a rectangular region $R$ in the $xy$-plane is given directly by $V = \iint_R f(x,y)\, dA$, provided $f(x,y) \geq 0$ on $R$.

Average value of a function over a region is

$f_{\text{avg}} = \frac{1}{\text{Area}(R)} \iint_R f(x,y)\, dA$

For a rectangle with dimensions $(b-a) \times (d-c)$, the area is simply $(b-a)(d-c)$. This comes up in heat transfer (average temperature over a plate), fluid mechanics (average pressure over a surface), and similar contexts where a single representative value over a region is needed.