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The Development of the Double Integral

Break the interval $[a,b]$ into $m$ pieces, each of size $\Delta x = \frac{ba}{m}$. Break the interval $[c,d]$ into $n$ pieces, each of size $\Delta y = \frac{dc}{n}$. This breaks $R$ into smaller rectangular boxes, which we call $R_{ij}$, where $i$ indicates the column and $j$ indicates the row. $R_{ij}$ has area $\Delta A =\Delta x\, \Delta y$. For each pair $i,j$, pick a sample point $\left(x_{ij}^*, y_{ij}^*\right)$ somewhere in $R_{ij}$. (For an approximation, a logical choice for the sample point $\left(x_{ij},y_{ij}\right)$ is given by the midpoint rule: $x_{ij}^* = a + \left(i\frac12\right)\,\Delta x, \qquad y_{ij}^* = c+\left(j\frac12\right)\Delta y.$) cnx.or

$xy$plane

Let the height of each tower over $R_{ij}$ be given by $f\left(x_{ij}^*,y_{ij}^*\right)$, i.e. by the height of the surface at our sample point over each rectangle. Find the volume of each tower over each rectangle $R_{ij}$ as $f\left(x_{ij}^*,y_{ij}^*\right) \,\Delta A$. Approximate the total volume by adding up all the tower volumes over all the $R_{ij}$, getting $$V\approx\displaystyle{\sum_{i=1}^m \sum_{j=1}^n f\left(x_{ij}^*, y_{ij}^*\right)\, \Delta A}.$$ www.stetson.edu

3space, with the "floor" being the $xy$plane 
In this example, you can see both the rectangles $R_{ij}$ on the $xy$plane, and above them the approximating columns, and the surface under which the volume is to be approximated. www.stetson.edu

With functions of one variable, we take the limit of the
approximations of area under a curve $f$ and get an
integral:$$A=\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^n
f(x_i)\Delta x.$$
Similarly, given our double sum approximating the volume, the limit of this sum is the definition of a (double) integral.
Definition: 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 $$\iint_R f(x,y)\, dA =
\iint_R f(x,y)\, dx \,dy = \lim_{m \to
\infty}\,\lim_{n\to\infty} \,\sum_{i=1}^m \sum_{j=1}^n
f\left(x_{ij}, y_{ij}\right)\, \Delta A.$$ (When we take the
limit, it doesn't matter which sample point we take, so we
take the upperright corner of the rectangle $R_{ij}$.) 