Knowlpedia

Fubini theorem for Riemann integrals

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

Let a<ba<b, c<dc<d, and let f:[a,b]×[c,d]→Rf:[a,b]\times[c,d]\to\mathbb R be continuous. Then ff is , both exist, and

∫ab ⁣(∫cdf(x,y) dy)dx=∫cd ⁣(∫abf(x,y) dx)dy=∬[a,b]×[c,d]f(x,y) dA.\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.
Remarks

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