Suppose we wish to compute [math]\int\int_Rf\left(x,y\right)dA[/math] over a rectangle [math]R=\left[a,b\right]\times\left[c,d\right][/math]. We could first compute [math]A\left(x\right)=\int_c^df\left(x,y\right)dy[/math], holding [math]x[/math] constant. Then, computing [math]\int_a^bA\left(x\right)dx[/math] gives us the answer. In other words, [br] [math]\int\int_Rf\left(x,y\right)dA=\int_a^b\left[\int_c^df\left(x,y\right)dy\right]dx[/math] which we just write as [math]\int_a^b\int_c^df\left(x,y\right)dydx[/math]. [br]This is called an [i]iterated integral[/i]. Fubini's Theorem tells us that if [math]f\left(x,y\right)[/math] is continuous throughout [math]R[/math], then [br] [math]\int\int_Rf\left(x,y\right)dA=\int_a^b\int_c^df\left(x,y\right)dydx=\int_c^d\int_a^bf\left(x,y\right)dxdy[/math].[br]That is, the order of integration does not matter. [br][br]In this interactive figure we demonstrate an iterated integral by first integrating with respect to [math]x[/math] then integrating with respect to [math]y[/math]. Click through each of the checkboxes in order and see why the iterated integral is a valid way of computing the double integral.[br]

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]