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Integrals over rectangles are
almost the same as integrals over intervals, except that now our
density $f(x,y)$ is the amount of stuff per unit - Break the interval $[a,b]$ into $m$ pieces, each of size $\Delta x = (b-a)/m$.
- Break the interval $[c,d]$ into $n$ pieces, each of size $\Delta y=(d-c)/n$.
- Together, this breaks $R$ into $nm$ smaller rectangular boxes, each of area $\Delta A =\Delta x \Delta y$. It breaks the solid into little towers of width $\Delta x$, depth $\Delta y$, and height $f(x,y)$.
- Label the boxes with two indices $i,j$, where $i$ says what column we're in and runs from 1 to $m$, while $j$ says what row we're in and runs from 1 to $n$.
- For each pair $i,j$, pick a
**sample point**$(x_{ij}^*, y_{ij}^*)$ somewhere in the $ij$-th box. - Approximate the volume of the tower over the $ij$-th box as $f(x_{ij}^*,y_{ij}^*) \Delta A$.
- Approximate the total volume as $\displaystyle{\sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, y_{ij}^*) \Delta A}$.
- To get an exact answer, take a limit as $n \to \infty$ and $m\to \infty$.
As with functions of one variable, the limit of this sum is the definition of a (double) integral.
In the following video, we develop the same definition with a slightly different problem, namely computing the total population of the (conveniently rectangular) state of Colorado. |