Applications of Double Integrals: Learn It 1

  • Use a double integral to calculate the area of a region, volume under a surface, or average value of a function over a plane region
  • Use double integrals to calculate the volume of a region between two surfaces or the area of a plane region
  • Solve problems involving improper double integrals

Applications of Double Integrals

Double integrals are very useful for finding the area of a region bounded by curves of functions. We describe this situation in more detail in the next section. However, if the region is a rectangular shape, we can find its area by integrating the constant function [latex]f(x,y) = 1[/latex] over the region [latex]R[/latex].

area of the region

The area of the region [latex]R[/latex] is given by:

[latex]A(R) = \underset{R}{\displaystyle\iint}1dA[/latex].

This definition makes sense because using [latex]f(x,y) = 1[/latex] and evaluating the integral make it a product of length and width. Let’s check this formula with an example and see how this works.

Find the area of the region [latex]{R} = {\left \{ (x,y) \mid 0 \leq x \leq 3,0 \leq y \leq 2 \right \}}[/latex] by using a double integral, that is, by integrating [latex]1[/latex] over the region [latex]R[/latex].

We have already seen how double integrals can be used to find the volume of a solid bounded above by a function [latex]f(x,y)[/latex] over a region [latex]R[/latex] provided [latex]f(x,y)\geq 0[/latex] for all [latex](x,y)[/latex] in [latex]R[/latex]. Here is another example to illustrate this concept.

Find the volume [latex]V[/latex] of the solid [latex]S[/latex] that is bounded by the elliptic paraboloid [latex]2x^2+y^2+z=27[/latex], the planes [latex]x = 3[/latex] and [latex]y = 3[/latex], and the three coordinate planes.

Similarly, we can define the average value of a function of two variables over a region [latex]R[/latex]. The main difference is that we divide by an area instead of the width of an interval.

Recall that we defined the average value of a function of one variable on an interval [latex][a,b][/latex] as:

[latex]{f_{\text{ave}}} = {\dfrac{1}{b-a}}\displaystyle\int_a^b{f(x)dx}[/latex].

average value of a function of two variables

The average value of a function of two variables over a region [latex]R[/latex] is:

[latex]{f_{\text{ave}}} = {\dfrac{1}{\text{Area }R}} \underset{R}{\displaystyle\iint}{f(x,y)dA}[/latex]

In the next example we find the average value of a function over a rectangular region. This is a good example of obtaining useful information for an integration by making individual measurements over a grid, instead of trying to find an algebraic expression for a function.

The weather map in Figure 2 shows an unusually moist storm system associated with the remnants of Hurricane Karl, which dumped 4–8 inches (100–200 mm) of rain in some parts of the Midwest on September 22–23, 2010. The area of rainfall measured 300 miles east to west and 250 miles north to south. Estimate the average rainfall over the entire area in those two days.

A map of Wisconsin and Minnesota that shows many cities with numbers affixed to them. The highest numbers come in a narrow band, and the map is colored accordingly. The map effectively looks like a contour map, but instead of elevations, it uses these numbers.
Figure 2. Effects of Hurricane Karl, which dumped 4–8 inches (100–200 mm) of rain in some parts of southwest Wisconsin, southern Minnesota, and southeast South Dakota over a span of 300 miles east to west and 250 miles north to south.