Fubini's Theorem

The reasoning of the previous pages can be summarized in a famous theorem:

Example:  Compute $\displaystyle\iint_R xe^y \,dA,$ where $R$ is the rectangle $[0,2] \times [0,1]$.

Solution:  We can integrate first over $x$. 
\begin{eqnarray}
\iint_R xe^y \, dA & = & \int_0^1 \left (\int_0^2 xe^y\, dx \right )\, dy\\
&=& \int_0^1 \left . \frac{x^2}{2} e^y \right |_{x=0}^2 \, dy \\
& = & \int_0^1 2 e^y\, dy \ = \ 2e-2
\end{eqnarray}

Another solution:  We can integrate first over $y$.  \begin{eqnarray} \iint_R xe^y \, dA & = & \int_0^2 \left (\int_0^1 xe^y \,dy \right )\, dx \cr \cr &=& \int_0^2 \left . x e^y \right|_{y=0}^1 \, dx \cr \cr & = & \int_0^2 (e-1)x\, dx \cr & = & \left . \frac{(e-1)x^2}{2} \right |_0^2 = 2e-2. \end{eqnarray}

Examples of unbounded and discontinuous functions $f(x,y)$ can be constructed where the iterated integral in one order is different from the iterated integral in the opposite order.   But for any reasonable function, you can integrate in either order.