Fubini theorem for Riemann integrals

For continuous functions on a rectangle, iterated integrals exist and agree with the double Riemann integral.

Fubini theorem for Riemann integrals

Fubini theorem (Riemann):

Let $a<b$ and $c<d$ , and let $f:[a,b]\times[c,d]\to\mathbb{R}$ be continuous. Then $f$ is Riemann integrable on the rectangle , the two iterated integrals exist, and

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

\iint_{[a,b]\times[c,d]} f(x,y)\,dA.

This allows one to compute a double Riemann integral by integrating one variable at a time, and it is a key tool for evaluating many multiple integrals in practice.