What is a double integral?

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 $m\to \infty$.

Investigate the interactive figure to the right. For each function, the integral of $f(x,y) dA$ is the volume of the region under the surface $z=f(x,y)$. The volume of each tower is $f(x_{ij}^*, y_{ij}^*) \Delta x \Delta y$. By varying the size of $\Delta x$ and $\Delta y$, and the choice of representative points $(x_{ij}^*, y_{ij}^*)$, you can see how the sums approximate the integral.

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: $x_{ij}^* = a + (i-\frac12)\Delta x$, $y_{ij}^* = c+(j-\frac12)\Delta y$.