Iterated Integrals and Fubini's Theorem

Iterated Integrals

We are about to look at a method of evaluating double integrals over rectangles and more general domains without using the definition of a double integral, but before we do so, we will first need to learn what an iterated integral.

Suppose that $z=f(x,y)$ is a two variable real-valued function, and suppose that $f$ is integrable for $a\leq x\leq b$ and $c\leq y\leq d$ , that is $f$ is integrable over the rectangle:

{\begin{aligned}\quad R=\{(x,y)\in \mathbb {R} ^{2}:a\leq x\leq b,c\leq y\leq d\}=[a,b]\times [c,d]\end{aligned}}

Just like partial differentiation with respect to a specific variable, we can also partial integrate with respect to a specific variable. For example, $\int _{c}^{d}f(x,y)\,dy$ means that we integrate $f$ with respect to $y$ while holding the variable $x$ as fixed. When we evaluate this integral, we will obtain a function in terms of $x$ only, and hence, we could then integrate the result from $a$ to $b$ with respect to $x$ as:

{\begin{aligned}\quad \int _{a}^{b}\int _{c}^{d}f(x,y)\,dy\,dx\end{aligned}}

The result above is what we call an iterated integral which we will define below.

Definition: If $z=f(x,y)$ is a two variable real valued function, and if $f$ is integrable over the rectangle $R=[a,b]\times [c,d]$ , then the Iterated Integral of $f$ over $R$ is

$\int _{a}^{b}\int _{c}^{d}f(x,y)\,dy\,dx$ .

We will see later that iterated integrals need not be over rectangles but instead can be done over more general domains.

One important property about iterated integrals is that we can partially integrate $f(x,y)$ with respect to either variable $x$ or $y$ first, and then continue onward with integrating with respect to the second variable, that is:

{\begin{aligned}\quad \int _{a}^{b}\int _{c}^{d}f(x,y)\,dy\,dx=\int _{c}^{d}\int _{a}^{b}f(x,y)\,dx\,dy\end{aligned}}

However, it is important to note that sometimes partial integrating with respect to a certain variable first will be a much easier process. Let's now look at some examples of evaluating iterated integrals.

Example 1

Evaluate the following iterated integral $\int _{2}^{4}\int _{1}^{3}x^{3}+xy^{2}\,dy\,dx$ . Over what rectangle is $f(x,y)=x^{3}+xy^{2}$ being integrated?

When evaluated iterated integrals over rectangles, we always want to work from the inside out. Let's first evaluate the inside integral with respect to $y$ while holding $x$ as fixed.

{\begin{aligned}\quad \int _{1}^{3}x^{3}+xy^{2}\,dy=\left[x^{3}y+{\frac {xy^{3}}{3}}\right]_{1}^{3}=\left(3x^{3}+9x\right)-\left(x^{3}+{\frac {x}{3}}\right)=2x^{3}+{\frac {26x}{3}}\end{aligned}}

Therefore we have that:

{\begin{aligned}\quad \int _{2}^{4}\int _{1}^{3}x^{3}+xy^{2}\,dy\,dx=\int _{2}^{4}2x^{3}+{\frac {26x}{3}}\,dx\end{aligned}}

Evaluating this definite integral and we get that:

{\begin{aligned}\quad \int _{2}^{4}2x^{3}+{\frac {26x}{3}}\,dx=\left[{\frac {x^{4}}{2}}+{\frac {26x^{2}}{6}}\right]_{2}^{4}=\left(128+{\frac {416}{6}}\right)-\left(8+{\frac {104}{26}}\right)=120+52=172\end{aligned}}

Therefore we have that:

{\begin{aligned}\quad \int _{2}^{4}\int _{1}^{3}x^{3}+xy^{2}\,dy\,dx=172\end{aligned}}

In this particular example, we are integrating over the rectangle $R=[2,4]\times [1,3]$ .

Example 2

Evaluate the following iterated integral $\int _{0}^{\pi }\int _{0}^{\pi }\sin x\cos y\,dx\,dy$ . Over what rectangle is $f(x,y)=\sin x\cos y$ being integrated?