Example: Find
Compute $$\iint_R xe^y \,dA,$$ where $R$ is the rectangle $[0,2]
\times [0,1]$.
Solution 1: 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 \cr \cr
&=& \int_0^1 \left . \frac{x^2}{2} e^y \right _{x=0}^2 \, dy \cr \cr
& = & \int_0^1 2 e^y\, dy \ = \ 2e2 \end{eqnarray*}
Solution 2: 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 (e1)x\, dx \cr & = &
\left . \frac{(e1)x^2}{2} \right _0^2 = 2e2.
\end{eqnarray*}
