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 area instead of stuff per unit length. The simplest example of `stuff' is volume, in which case $f(x,y)$ is height. So suppose that $R = [a,b] \times [c,d]$ is a rectangle, where $x$ runs from $a$ to $b$ and $y$ runs from $c$ to $d$. Let's figure out the volume of the solid between the $x$-$y$ plane and the surface $z=f(x,y)$, and over the rectangle $R$.

  1. Break the interval $[a,b]$ into $m$ pieces, each of size $\Delta x = (b-a)/m$.
  2. Break the interval $[c,d]$ into $n$ pieces, each of size $\Delta y=(d-c)/n$.
  3. 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 $f(x,y)$.
  4. 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 $n$.
  5. For each pair $i,j$, pick a sample point $(x_{ij}^*, y_{ij}^*)$ somewhere in the $ij$-th box.
  6. Approximate the volume of the tower over the $ij$-th box as $f(x_{ij}^*,y_{ij}^*) \Delta A$.
  7. Approximate the total volume as $\displaystyle{\sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, y_{ij}^*) \Delta A}$.
  8. To get an exact answer, take a limit as $n \to \infty$ and $m\to \infty$.

The figure to the right shows this process for the function $f(x,y)=x^2+y^2$, where $R = [-10,10] \times [-10,10]$, and where $n=m=8$. The volume under the surface is approximately the sum of the volumes of a number of towers, and the volume of each tower is $f(x_{ij}^*, y_{ij}^*) \Delta x \Delta y$. As $n \to \infty$ and $m \to \infty$, the approximation gets better and better, and in the limit gives the exact volume under the surface.

As with functions of one variable, the limit of this sum is the definition of a (double) integral.

If $f(x,y)$ is a continuous function of two variables and if $R = [a,b] \times [c,d]$ is a rectangle, then the double integral of $f$ over $R$ is $$\iint_R f(x,y) dA = \iint_R f(x,y) dx dy = \lim_{m \to \infty}\lim_{n\to\infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, y_{ij}^*) \Delta A.$$ A convenient choice for the sample point $(x_{ij}^*,y_{ij}^*)$ is given by the midpoint rule: $x_{ij}^* = a + (i-\frac12)\Delta x$, $y_{ij}^* = c+(j-\frac12)\Delta y$.


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.