|
Fubini's theorem Let
 and
 be compact intervals, and let
 be a Riemann integrable function such that, for each  the integral
exists. Then
 is Riemann integrable, and
This theorem effectively states that, given a function of  variables, you may integrate it one variable at a time, and that the order of integration does not affect the result.
Example Let
![$ I := [0, \pi/2]\times[0,\pi/2]$](http://images.planetmath.org:8080/cache/objects/4307/l2h/img9.png "$ I := [0, \pi/2]\times[0,\pi/2]$") , and let
 be a function. Then
Note that it is often simpler (and no less correct) to write
 as  .
| 
![\begin{displaymath}\begin{split}\int_I f &= \iint_{[0, \pi/2]\times[0,\pi/2]} \s... ...\pi/2} \sin(x)\left(1 - 0\right)\,dx =(0 - -1) = 1. \end{split}\end{displaymath}](http://images.planetmath.org:8080/cache/objects/4307/l2h/img11.png "\begin{displaymath}\begin{split}\int_I f &= \iint_{[0, \pi/2]\times[0,\pi/2]} \s... ...\pi/2} \sin(x)\left(1 - 0\right)\,dx =(0 - -1) = 1. \end{split}\end{displaymath}")