#uw/notes #uw/class/math224 
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Recall that:
$$\int \int_{R} f(x, y) dA$$
is the signed area under the function $f(x,y)$ over the domain $R$ where 
$$R=[a,b] \times [c,d]$$
and it can be written as 
$$\int_{a}^b \int_{c}^d f(x, y) dy dx$$

This can be extended to more general domains that are not rectangles such as $[a,b] \times [c,d]$

Example:
finding the volume under the function $f(x,y) = xy$ over the domain bound by 
$$\begin{gathered} 1<x<2 \\[1.5ex] x < y < x^2 \end{gathered}$$
We can think of finding the volume of the solid by using slices parallel to the $YZ$ plane. With the domain provided this area can be found with 
$$ Area(x^*)=\int_{x}^{x^2}x^*ydy$$
Since $x$ is bound by $1<x<2$ the double integral to find the volume can be written as 
$$\begin{gather} \int_{1}^2 \left[ \int_{x}^{x^2}xy \  dy \right] dx \\[1.5ex] \int_{1}^2 \left[ \int_{x}^{x^2} \frac{xy^2}{2}   \right] dx \end{gather}$$