Multiple Integrals

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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 $xy$ plane and the surface $z=f(x,y)$, and over the rectangle $R$.

Break the interval $[a,b]$ into $m$ pieces, each of size $\displaystyle\Delta x = \frac{b-a}{m}$.

Break the interval $[c,d]$ into $n$ pieces, each of size $\displaystyle\Delta y=\frac{d-c}{n}$.

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 $\left(x_{ij}^*, y_{ij}^*\right)$ somewhere in the $ij$-th box.

Approximate the volume of the tower over the $ij$-th box as $f\left(x_{ij}^*,y_{ij}^*\right) \,\Delta A$.

Approximate the total volume as $$\displaystyle{\sum_{i=1}^m \sum_{j=1}^n f\left(x_{ij}^*, y_{ij}^*\right)\, \Delta A}.$$
To get an exact answer, take a limit as $n \to \infty$ and $m\to \infty$.

Interactive:

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 $f\left(x_{ij}^*, y_{ij}^*\right) \Delta x \Delta y$. By varying the size of $\Delta x$ and $\Delta y$, and the choice of representative points $\left(x_{ij}^*, y_{ij}^*\right)$, 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 $$\iint_R f(x,y)\, dA = \iint_R f(x,y)\, dx \,dy = \lim_{m \to \infty}\,\lim_{n\to\infty} \,\sum_{i=1}^m \sum_{j=1}^n f\left(x_{ij}^*, y_{ij}^*\right)\, \Delta A.$$
A convenient choice for the sample point $\left(x_{ij}^*,y_{ij}^*\right)$ is given by the midpoint rule: $$x_{ij}^* = a + \left(i-\frac12\right)\,\Delta x, \qquad y_{ij}^* = c+\left(j-\frac12\right)\Delta y.$$

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.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

What is a double integral?