Iterated integral - Knowlpedia

An iterated integral of $f:[a,b]\times[c,d]\to\mathbb R$ integrates in one variable and then the other. The $y$ -then- $x$ iterated integral is

\int_a^b\left(\int_c^d f(x,y)\,dy\right)dx,

provided that the inner Riemann integral exists for every $x\in[a,b]$ and the resulting function of $x$ is Riemann integrable.

Iterated integrals are compared with the multiple Riemann integral; under appropriate hypotheses they agree by Fubini's theorem (Riemann).

Examples

For $f(x,y)=xy$ on $[0,1]^2$ , the iterated integral is $1/4$ .
If $f(x,y)=g(x)h(y)$ , then the iterated integral equals $\left(\int_a^b g(x)\,dx\right)\left(\int_c^d h(y)\,dy\right)$ whenever the one-variable integrals exist.