Iterated integral

Let $R=[a,b]\times[c,d]\subseteq \mathbb{R}^2$ and let $f:R\to\mathbb{R}$ be a function such that for each fixed $x\in[a,b]$ the function $y\mapsto f(x,y)$ is (Riemann) integrable on $[c,d]$, and similarly for each fixed $y$.

The iterated integrals are $\int_a^b\left(\int_c^d f(x,y)\,dy\right)dx \quad\text{and}\quad \int_c^d\left(\int_a^b f(x,y)\,dx\right)dy,$ provided the inner integrals exist and the resulting outer integrals exist.

Iterated integrals are computationally convenient; Fubini-type theorems give hypotheses under which iterated integrals agree with the genuine multiple integral .

Examples:

• If $f(x,y)=x+y$ on $[0,1]^2$, then $\int_0^1\!\int_0^1 (x+y)\,dy\,dx=1$.
• For continuous $f$ on a rectangle, both iterated integrals exist and agree with the double integral.