Fubini's Theorem

Fubini’s theorem allows swapping the variables being integrated with respect to, provided the integrated function is continuous. It is extremely useful in computing double integrals and triple integrals.

Rectangular-like Domain §

Fubini's Theorem (Rectangular Domain) ^fubini-theorem-rect

Let $f:{\mathbb{R}}^2\to \mathbb{R}$ be a continuous bivariate function over a rectangular domain $R=[a,b]\times [c,d]$. Then:

$${\iint }_{R}f(x,y)\text{d}A={\int }_c^d{\int }_a^{b}f(x,y)\text{d}x\text{d}y=\int \int f(x,y)\text{d}y\text{d}x$$

Fubini's Theorem (Cuboidal Domain) ^fubini-theorem-cuboid

Let $f:{\mathbb{R}}^3\to \mathbb{R}$ be a continuous trivariate function over a cuboidal domain $C=[a,b]\times [c,d]\times [e,f]$. Then:

\begin{align} \iint _{C} f(x,y,z) \, \text{d}{V} &= \int_{e}^f \int_{c}^d \int _{a}^b f(x,y,z) \, \text{d}{x} \, \text{d}{y} \, \text{d}{z} \\ &= \int_{c}^d \int_{e}^f \int _{a}^b f(x,y,z) \, \text{d}{x} \, \text{d}{z} \, \text{d}{y} \\

&= \dots (\text{Any permutation of } \left( \int_{a}^b f , \text{d}x, \int_{c}^d f , \text{d}y, \int_{e}^f f , \text{d}z \right))

\end{align}

Fubini's Theorem (Hypercube Domain) ^fubini-theorem-hypercube

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a continuous multivariate function over a $n$-dimensional hypercube $C=[a_1,b_1]\times [a_2,b_2]\times \cdots \times [a_n,b_n]$. Then:

$$\int \cdots {\int }_{C}f(x_1,\dots ,x_n)\text{d}{x}_1\dots \text{d}{x}_n=\text{Any permutation of integrations w.r.t x1…xn}$$

i.e. the order of integration does not matter, as long as we’re applying the correct bounds.

General Domain §

The rectangular case can be extended to obtain a more general version:

Fubini's Theorem ^fubini-theorem-general

Let $f:{\mathbb{R}}^2\to \mathbb{R}$ be a continuous bivariate function over a domain $D$. Then, if $D$ is both horizontally simple and vertically simple, we can swap the variables:

$${\iint }_{D}f(x,y)\text{d}A={\iint }_{D}f(x,y)\text{d}x\text{dy}={\iint }_{D}f(x,y)\text{d}y\text{d}x$$

I.e. the order of integration does not matter.

Product Of Integrals §

A useful corollary of Fubini’s theorem is that the integral of products is the products of integrals.

C1: Integral of products is the product of integrals ^c1

Let $f$#todo

Derivation §

Take a function $f(x,y)$ integrated over some domain $D=[a,b]\times [c,d]$. Assume that this function can be factorised with respect to each variable, that is, $f(x,y)=g(x)\cdot h(y)$. Then:

$${\iint }_{D}f(x,y)\text{d}x\text{d}y={\int }_c^d{\int }_a^{b}g(x)\cdot h(y)\text{d}x\text{d}y={\int }_c^d\left({\int }_a^{b}g(x)\cdot h(y)\text{d}x\right)\text{d}y$$

We can ‘pull out’ the $h(y)$, since it is a constant with respect to $x$:

$${\int }_c^d\left({\int }_a^{b}g(x)\cdot h(y)\text{d}x\right)\text{d}y={\int }_c^{d}h(y)\cdot \left({\int }_a^{b}g(x)\text{d}x\right)\text{d}y$$

But then, with respect to $y$, the integral ${\int }_a^{b}g(x)\text{d}x$ only contains $x$ terms and constants, so is constant w.r.t $y$ and can also be pulled out:

$${\int }_c^{d}h(y)\cdot \left({\int }_a^{b}g(x)\text{d}x\right)\text{d}y={\int }_a^{b}g(x)\text{d}x\cdot {\int }_c^{d}h(y)\text{d}y$$

The reason we need Fubini’s theorem is that it allows swapping the two integrals, which means it obeys the commutativity of multiplication.