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3.3: Double Integrals Over General Regions

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Learning Objectives
  • Recognize when a function of two variables is integrable over a general region.
  • Evaluate a double integral by computing an iterated integral over a region bounded by two vertical lines and two functions of x, or two horizontal lines and two functions of y.
  • Simplify the calculation of an iterated integral by changing the order of integration.
  • Use double integrals to calculate the volume of a region between two surfaces or the area of a plane region.
  • Solve problems involving double improper integrals.

Previously, we studied the concept of double integrals and examined the tools needed to compute them. We learned techniques and properties to integrate functions of two variables over rectangular regions. We also discussed several applications, such as finding the volume bounded above by a function over a rectangular region, finding area by integration, and calculating the average value of a function of two variables.

In this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in this situation as well, but some techniques need to be extended to cover this more general case.

General Regions of Integration

An example of a general bounded region D on a plane is shown in Figure 3.3.1. Since D is bounded on the plane, there must exist a rectangular region R on the same plane that encloses the region D that is, a rectangular region R exists such that D is a subset of R(DR).

A rectangle R with a shape D inside of it. Inside D, there is a point labeled g(x, y) = f(x, y). Outside D but still inside R, there is a point labeled g(x, y) = 0.
Figure 3.3.1: For a region D that is a subset of R, we can define a function g(x,y) to equal f(x,y) at every point in D and 0 at every point of R not in D.

Suppose z=f(x,y) is defined on a general planar bounded region D as in Figure 3.3.1. In order to develop double integrals of f over D we extend the definition of the function to include all points on the rectangular region R and then use the concepts and tools from the preceding section. But how do we extend the definition of f to include all the points on R? We do this by defining a new function g(x,y) on R as follows:

g(x,y)={f(x,y),if(x,y)is inD0,if(x,y)is inRbut not inD

Note that we might have some technical difficulties if the boundary of D is complicated. So we assume the boundary to be a piecewise smooth and continuous simple closed curve. Also, since all the results developed in the section on Double Integrals over Rectangular Regions used an integrable function f(x,y) we must be careful about g(x,y) and verify that g(x,y) is an integrable function over the rectangular region R. This happens as long as the region D is bounded by simple closed curves. For now we will concentrate on the descriptions of the regions rather than the function and extend our theory appropriately for integration.

We consider two types of planar bounded regions.

Definition: Type I and Type II regions

A region D in the (x,y)-plane is of Type I if it lies between two vertical lines and the graphs of two continuous functions g1(x) and g2(x). That is (Figure 3.3.2),

D={(x,y)|axb, g1(x)yg2(x)}.

A region D in the xy-plane is of Type II if it lies between two horizontal lines and the graphs of two continuous functions h1(y) and h2(y). That is (Figure 3.3.3),

D={(x,y)|cyd, h1(y)xh2(y)}.

The graphs showing a region marked D. In all instances, between a and b, there is a shape that is defined by two functions g1(x) and g2(x). In one instance, the two functions do not touch; in another instance, they touch at the end point a, and in the last instance they touch at both end points.
Figure 3.3.2. A Type I region lies between two vertical lines and the graphs of two functions of x.
The graphs show a region marked D. In all instances, between c and d, there is a shape that is defined by two vertically oriented functions x = h1(y) and x = h2(y). In one instance, the two functions do not touch; in the other instance, they touch at the end point c.
Figure 3.3.3: A Type II region lies between two horizontal lines and the graphs of two functions of y.
Example 3.3.1: Describing a Region as Type I and Also as Type II

Consider the region in the first quadrant between the functions y=x and y=x3 (Figure 3.3.4). Describe the region first as Type I and then as Type II.

The region D is drawn between two functions, namely, y = the square root of x and y = x3.
Figure 3.3.4: Region D can be described as Type I or as Type II.

When describing a region as Type I, we need to identify the function that lies above the region and the function that lies below the region. Here, region D is bounded above by y=x and below by y=x3 in the interval for x in [0,1]. Hence, as Type I, D is described as the set {(x,y)|0x1, x3y3x}.

However, when describing a region as Type II, we need to identify the function that lies on the left of the region and the function that lies on the right of the region. Here, the region D is bounded on the left by x=y2 and on the right by x=3y in the interval for y in [0,1]. Hence, as Type II, D is described as the set {(x,y)|0y1, y2x3y}.

Exercise 3.3.1

Consider the region in the first quadrant between the functions y=2x and y=x2. Describe the region first as Type I and then as Type II.

Hint

Graph the functions, and draw vertical and horizontal lines.

Answer

Type I and Type II are expressed as {(x,y)|0x2, x2y2x} and {(x,y)|0y4, 12yxy}, respectively.

Double Integrals over Non-rectangular Regions

To develop the concept and tools for evaluation of a double integral over a general, nonrectangular region, we need to first understand the region and be able to express it as Type I or Type II or a combination of both. Without understanding the regions, we will not be able to decide the limits of integrations in double integrals. As a first step, let us look at the following theorem.

Theorem: Double Integrals over Nonrectangular Regions

Suppose g(x,y) is the extension to the rectangle R of the function f(x,y) defined on the regions D and R as shown in Figure 3.3.1 inside R. Then g(x,y) is integrable and we define the double integral of f(x,y) over D by

The right-hand side of this equation is what we have seen before, so this theorem is reasonable because R is a rectangle and \iint\limits_R g(x,y)dA has been discussed in the preceding section. Also, the equality works because the values of g(x,y) are 0 for any point (x,y) that lies outside D and hence these points do not add anything to the integral. However, it is important that the rectangle R contains the region D.

As a matter of fact, if the region D is bounded by smooth curves on a plane and we are able to describe it as Type I or Type II or a mix of both, then we can use the following theorem and not have to find a rectangle R containing the region.

Theorem: Fubini’s Theorem (Strong Form)

For a function f(x,y) that is continuous on a region D of Type I, we have

\iint\limits_D f(x,y)\,dA = \iint\limits_D f(x,y)\,dy \space dx = \int_a^b \left[\int_{g_1(x)}^{g_2(x)} f(x,y)\,dy \right] dx. \nonumber

Similarly, for a function f(x,y) that is continuous on a region D of Type II, we have

\iint\limits_D f(x,y)\,dA = \iint\limits_D f(x,y)\,dx \space dy = \int_c^d \left[\int_{h_1(y)}^{h_2(y)} f(x,y)\,dx \right] dy. \nonumber

The integral in each of these expressions is an iterated integral, similar to those we have seen before. Notice that, in the inner integral in the first expression, we integrate f(x,y) with x being held constant and the limits of integration being g_1(x) and g_2(x). In the inner integral in the second expression, we integrate f(x,y) with y being held constant and the limits of integration are h_1(x) and h_2(x).

Example \PageIndex{2}: Evaluating an Iterated Integral over a Type I Region

Evaluate the integral \displaystyle \iint \limits _D x^2 e^{xy} \,dA where D is shown in Figure \PageIndex{5}.

Solution

First construct the region as a Type I region (Figure \PageIndex{5}). Here D = \big\{(x,y) \,|\, 0 \leq x \leq 2, \space \frac{1}{2} x \leq y \leq 1\big\}. Then we have

\iint \limits _D x^2e^{xy} \,dA = \int_{x=0}^{x=2} \int_{y=1/2x}^{y=1} x^2e^{xy}\,dy\,dx. \nonumber

A triangle marked D drawn with lines x = 2y and y = 1, with vertices (0, 0), (2, 1), and (0, 1). Here there is a pair of red arrows reaching horizontally from one edge to the other.
Figure \PageIndex{5}: We can express region D as a Type I region and integrate from y = \frac{1}{2} x to y = 1 between the lines x = 0 and x = 2.

Therefore, we have

\begin{align*} \int_{x=0}^{x=2}\int_{y=\frac{1}{2}x}^{y=1}x^2e^{xy}\,dy\,dx &= \int_{x=0}^{x=2}\left[\int_{y=\frac{1}{2}x}^{y=1}x^2e^{xy}\,dy\right] dx & &\text{Iterated integral for a Type I region.}\\[5pt] &=\int_{x=0}^{x=2} \left.\left[ x^2 \frac{e^{xy}}{x} \right] \right|_{y=1/2x}^{y=1}\,dx & & \text{Integrate with respect to $y$}\\[5pt] &= \int_{x=0}^{x=2} \left[xe^x - xe^{x^2/2}\right]dx & & \text{Integrate with respect to $x$} \\[5pt] &=\left[xe^x - e^x - e^{\frac{1}{2}x^2} \right] \Big|_{x=0}^{x=2} = 2. \end{align*}

In Example \PageIndex{2}, we could have looked at the region in another way, such as D = \big\{(x,y)\,|\,0 \leq y \leq 1, \space 0 \leq x \leq 2y\big\} (Figure \PageIndex{6}).

alt
Figure \PageIndex{6}.

This is a Type II region and the integral would then look like

\iint \limits _D x^2e^{xy}\,dA = \int_{y=0}^{y=1} \int_{x=0}^{x=2y} x^2 e^{xy}\,dx \space dy. \nonumber

However, if we integrate first with respect to x this integral is lengthy to compute because we have to use integration by parts twice.

Example \PageIndex{3}: Evaluating an Iterated Integral over a Type II Region

Evaluate the integral

\iint \limits _D (3x^2 + y^2) \,dA \nonumber

where D = \big\{(x,y)\,| \, -2 \leq y \leq 3, \space y^2 - 3 \leq x \leq y + 3\big\}.

Solution

Notice that D can be seen as either a Type I or a Type II region, as shown in Figure \PageIndex{7}. However, in this case describing D as Type I is more complicated than describing it as Type II. Therefore, we use D as a Type II region for the integration.

This figure consists of two figures labeled a and b. In figure a, a region is bounded by y = the square root of the quantity (x + 3), y = the negative of the square root of the quantity (x + 3), and y = x minus 3, which has points of intersection (6, 3), (1, negative 2), and (0, negative 3). There are vertical lines in the shape, and it is noted that this is a type I region: integrate first with respect to y. In figure b, a region is bounded by x = y2 minus 3 and x = y + 3, which has points of intersection (6, 3), (1, negative 2), and (0, negative 3). There are horizontal lines in the shape, and it is noted that this is a type II region: integrate first with respect to x.
Figure \PageIndex{7}: The region D in this example can be either (a) Type I or (b) Type II.

Choosing this order of integration, we have

\begin{align*} \iint \limits _D (3x^2 + y^2)\,dA &= \int_{y=-2}^{y=3} \int_{x=y^2-3}^{x=y+3} (3x^2 + y^2) \,dx \space dy \\[5pt] &=\int_{y=-2}^{y=3} \left. (x^3 + xy^2) \right|_{y^2-3}^{y+3} \,dy & & \text{Iterated integral, Type II region}\\[5pt] &=\int_{y=-2}^{y=3} \left((y + 3)^3 + (y + 3)y^2 - (y^2 - 3)y^2\right)\,dy \\ &=\int_{-2}^3 (54 + 27y - 12y^2 + 2y^3 + 8y^4 - y^6)\,dy & & \text{Integrate with respect to $x$.} \\[5pt] &= \left[ 54y + \frac{27y^2}{2} - 4y^3 + \frac{y^4}{2} + \frac{8y^5}{5} - \frac{y^7}{7} \right]_{-2}^3 \\ &=\frac{2375}{7}. \end{align*}

Exercise \PageIndex{2}

Sketch the region D and evaluate the iterated integral \iint \limits _D xy \space dy \space dx \nonumber where D is the region bounded by the curves y = \cos \space x and y = \sin \space x in the interval [-3\pi/4, \space \pi/4].

Hint

Express D as a Type I region, and integrate with respect to y first.

Answer

\frac{\pi}{4}

Recall from Double Integrals over Rectangular Regions the properties of double integrals. As we have seen from the examples here, all these properties are also valid for a function defined on a non-rectangular bounded region on a plane. In particular, property 3 states:

If R = S \cup T and S \cap T = 0 except at their boundaries, then

\iint \limits _R f(x,y)\,dA = \iint\limits _S f(x,y)\,dA + \iint\limits _T f(x,y) \,dA. \nonumber

Similarly, we have the following property of double integrals over a non-rectangular bounded region on a plane.

Theorem: Decomposing Regions into Smaller Regions

Suppose the region D can be expressed as D = D_1 \cup D_2 where D_1 and D_2 do not overlap except at their boundaries. Then

\iint \limits _D f(x,y) \,dA = \iint \limits _{D_1} f(x,y) \,dA + \iint \limits _{D_2} f(x,y) \,dA. \nonumber

This theorem is particularly useful for non-rectangular regions because it allows us to split a region into a union of regions of Type I and Type II. Then we can compute the double integral on each piece in a convenient way, as in the next example.

Example \PageIndex{4}: Decomposing Regions

Express the region D shown in Figure \PageIndex{8} as a union of regions of Type I or Type II, and evaluate the integral

\iint \limits _D (2x + 5y)\,dA. \nonumber

A complicated shape enclosed by the lines y = (x + 2) squared, x = 16y minus y cubed, x = negative 2, and y = negative 4. This graph has intersection points (0, 4), (negative 2, 0), (0, negative 4), and (negative 2, negative 4).
Figure \PageIndex{8}: This region can be decomposed into a union of three regions of Type I or Type II.
Solution

The region D is not easy to decompose into any one type; it is actually a combination of different types. So we can write it as a union of three regions D_1, D_2, and D_3 where, D_1 = \big\{(x,y)\,| \, -2 \leq x \leq 0, \space 0 \leq y \leq (x + 2)^2 \big\}, D_2 = \big\{(x,y)\,| \, 0 \leq y \leq 4, \space 0 \leq x \leq \big(y - \frac{1}{16} y^3 \big) \big\}, and D_3 = \big\{(x,y)\,| \, -4 \leq y \leq 0, \space -2 \leq x \leq \big(y - \frac{1}{16} y^3 \big) \big\}. These regions are illustrated more clearly in Figure \PageIndex{9}.

The same complicated shape enclosed by the lines y = (x + 2) squared, x = 16y minus y cubed, x = negative 2, and y = negative 4. This graph has intersection points (0, 4), (negative 2, 0), (0, negative 4), and (negative 2, negative 4). The area in the first quadrant is marked as D2 and a Type II region. The region in the second quadrant is marked as D1 and is a Type I region. The region in the third quadrant is marked as D3 and is a Type II region.
Figure \PageIndex{9}: Breaking the region into three subregions makes it easier to set up the integration.

Here D_1 is Type I and D_2 and D_3 are both of Type II. Hence,

\begin{align*} \iint\limits_D (2x + 5y)\,dA &= \iint\limits_{D_1} (2x + 5y)\,dA + \iint\limits_{D_2} (2x + 5y)\,dA + \iint\limits_{D_3} (2x + 5y)\,dA \\ &= \int_{x=-2}^{x=0} \int_{y=0}^{y=(x+2)^2} (2x + 5y) \,dy \space dx + \int_{y=0}^{y=4} \int_{x=0}^{x=y-(1/16)y^3} (2 + 5y)\,dx \space dy + \int_{y=-4}^{y=0} \int_{x=-2}^{x=y-(1/16)y^3} (2x + 5y)\,dx \space dy \\ &= \int_{x=-2}^{x=0} \left[\frac{1}{2}(2 + x)^2 (20 + 24x + 5x^2)\right]\,dx + \int_{y=0}^{y=4} \left[\frac{1}{256}y^6 - \frac{7}{16}y^4 + 6y^2 \right]\,dy +\int_{y=-4}^{y=0} \left[\frac{1}{256}y^6 - \frac{7}{16}y^4 + 6y^2 + 10y - 4\right] \,dy\\ &= \frac{40}{3} + \frac{1664}{35} - \frac{1696}{35} = \frac{1304}{105}.\end{align*}

Now we could redo this example using a union of two Type II regions (see the Checkpoint).

Exercise \PageIndex{3}

Consider the region bounded by the curves y = \ln x and y = e^x in the interval [1,2]. Decompose the region into smaller regions of Type II.

Hint

Sketch the region, and split it into three regions to set it up.

Answer

\big\{(x,y)\,| \, 0 \leq y \leq 1, \space 1 \leq x \leq e^y \big\} \cup \big\{(x,y)\,| \, 1 \leq y \leq e, \space 1 \leq x \leq 2 \big\} \cup \big\{(x,y)\,| \, e \leq y \leq e^2, \space \ln y \leq x \leq 2 \big\} \nonumber

Exercise \PageIndex{4}

Redo Example \PageIndex{4} using a union of two Type II regions.

Hint

\big\{(x,y)\,| \, 0 \leq y \leq 4, \space 2 + \sqrt{y} \leq x \leq \big(y - \frac{1}{16}y^3\big) \big\} \cup \big\{(x,y)\,| \, - 4 \leq y \leq 0, \space -2 \leq x \leq \big(y - \frac{1}{16}y^{13}\big) \big\} \nonumber

Answer

Same as in the example shown.

Changing the Order of Integration

As we have already seen when we evaluate an iterated integral, sometimes one order of integration leads to a computation that is significantly simpler than the other order of integration. Sometimes the order of integration does not matter, but it is important to learn to recognize when a change in order will simplify our work.

Example \PageIndex{5}: Changing the Order of Integration

Reverse the order of integration in the iterated integral

\int_{x=0}^{x=\sqrt{2}} \int_{y=0}^{y=2-x^2} xe^{x^2} \,dy \space dx. \nonumber

Then evaluate the new iterated integral.

Solution

The region as presented is of Type I. To reverse the order of integration, we must first express the region as Type II. Refer to Figure \PageIndex{10}.

This figure consists of two figures labeled Type I and Type II. In the Type I figure, a curve is given as y = 2 minus x squared, which forms a shape with the x and y axes. There is a vertical line with arrows on the end of it within this shape. In the Type II figure, a curve is given as x = the square root of the quantity (2 minus y), which forms a shape with the x and y axes. There is a horizontal line with arrows on the end of it within this shape.
Figure \PageIndex{10}: Converting a region from Type I to Type II.

We can see from the limits of integration that the region is bounded above by y = 2 - x^2 and below by y = 0 where x is in the interval [0, \sqrt{2}]. By reversing the order, we have the region bounded on the left by x = 0 and on the right by x = \sqrt{2 - y} where y is in the interval [0, 2]. We solved y = 2 - x^2 in terms of x to obtain x = \sqrt{2 - y}.

Hence

\begin{align*} \int_0^{\sqrt{2}} \int_0^{2-x^2} xe^{x^2} dy \space dx &= \int_0^2 \int_0^{\sqrt{2-y}} xe^{x^2}\,dx \space dy &\text{Reverse the order of integration then use substitution.} \\[4pt] &= \int_0^2 \left[\left.\frac{1}{2}e^{x^2}\right|_0^{\sqrt{2-y}}\right] dy = \int_0^2\frac{1}{2}(e^{2-y} - 1)\,dy \\[4pt] &= -\left.\frac{1}{2}(e^{2-y} + y)\right|_0^2 = \frac{1}{2}(e^2 - 3). \end{align*}

Example \PageIndex{6}: Evaluating an Iterated Integral by Reversing the Order of Integration

Consider the iterated integral

\iint\limits_R f(x,y)\,dx \space dy \nonumber

where z = f(x,y) = x - 2y over a triangular region R that has sides on x = 0, \space y = 0, and the line x + y = 1. Sketch the region, and then evaluate the iterated integral by

  1. integrating first with respect to y and then
  2. integrating first with respect to x.
Solution

A sketch of the region appears in Figure \PageIndex{11}.

The line y = 1 minus x is drawn, and it is also marked as x = 1 minus y. There is a shaded region around x = 0 that comes from the y axis, which projects down to make a shaded region marked y = 0 from the x axis.
Figure \PageIndex{11}: A triangular region R for integrating in two ways.

We can complete this integration in two different ways.

a. One way to look at it is by first integrating y from y = 0 to y = 1 - x vertically and then integrating x from x = 0 to x = 1:

\begin{align*} \iint\limits_R f(x,y) \,dx \space dy &= \int_{x=0}^{x=1} \int_{y=0}^{y=1-x} (x - 2y) \,dy \space dx = \int_{x=0}^{x=1}\left(xy - 2y^2\right)\Big|_{y=0}^{y=1-x} dx \\[4pt] &=\int_{x=0}^{x=1} \left[ x(1 - x) - (1 - x)^2\right] \,dx = \int_{x=0}^{x=1} [ -1 + 3x - 2x^2] dx = \left[ -x + \frac{3}{2}x^2 - \frac{2}{3} x^3 \right]\Big|_{x=0}^{x=1} = -\frac{1}{6}. \end{align*}

b. The other way to do this problem is by first integrating x from x = 0 to x = 1 - y horizontally and then integrating y from y = 0 to y = 1:

\begin{align*} \iint \limits _D (3x^2 + y^2)\,dA &= \int_{y=-2}^{y=3} \int_{x=y^2-3}^{x=y+3} (3x^2 + y^2) \,dx \space dy \\[4pt] &=\int_{y=-2}^{y=3} (x^3 + xy^2) \Big|_{y^2-3}^{y+3} \,dy & & \text{Iterated integral, Type II region}\\[4pt] &=\int_{y=-2}^{y=3} \left((y + 3)^3 + (y + 3)y^2 - (y^2 - 3)y^2\right)\,dy \\[4pt] &=\int_{-2}^3 (54 + 27y - 12y^2 + 2y^3 + 8y^4 - y^6)\,dy & & \text{Integrate with respect to $x$.} \\[4pt] &= \left( 54y + \frac{27y^2}{2} - 4y^3 + \frac{y^4}{2} + \frac{8y^5}{5} - \frac{y^7}{7} \right)\Big|_{-2}^3 \\[4pt] &=\frac{2375}{7}. \end{align*}

Exercise \PageIndex{5}

Evaluate the iterated integral \displaystyle \iint\limits_D (x^2 + y^2)\,dA over the region D in the first quadrant between the functions y = 2x and y = x^2. Evaluate the iterated integral by integrating first with respect to y and then integrating first with resect to x.

Hint

Sketch the region and follow Example \PageIndex{6}.

Answer

\frac{216}{35}

Calculating Volumes, Areas, and Average Values

We can use double integrals over general regions to compute volumes, areas, and average values. The methods are the same as those in Double Integrals over Rectangular Regions, but without the restriction to a rectangular region, we can now solve a wider variety of problems.

Example \PageIndex{7}: Finding the Volume of a Tetrahedron

Find the volume of the solid bounded by the planes x = 0, \space y = 0, \space z = 0, and 2x + 3y + z = 6.

Solution

The solid is a tetrahedron with the base on the xy-plane and a height z = 6 - 2x - 3y. The base is the region D bounded by the lines, x = 0, y = 0 and 2x + 3y = 6 where z = 0 (Figure \PageIndex{12}). Note that we can consider the region D as Type I or as Type II, and we can integrate in both ways.

This figure shows a tetrahedron bounded by x = 0, y = 0, z = 0, and 2x + 3y = 6 (or z = 6 minus 2x minus 3y).
Figure \PageIndex{12}: A tetrahedron consisting of the three coordinate planes and the plane z = 6 - 2x - 3y, with the base bound by x = 0, \space y = 0, and 2x + 3y = 6.

First, consider D as a Type I region, and hence D = \big\{(x,y)\,| \, 0 \leq x \leq 3, \space 0 \leq y \leq 2 - \frac{2}{3} x \big\}.

Therefore, the volume is

\begin{align*} V &= \int_{x=0}^{x=3} \int_{y=0}^{y=2-(2x/3)} (6 - 2x - 3y) \,dy \space dx = \int_{x=0}^{x=3} \left[ \left.\left( 6y - 2xy - \frac{3}{2}y^2\right)\right|_{y=0}^{y=2-(2x/3)} \right] \,dx\\[4pt] &= \int_{x=0}^{x=3} \left[\frac{2}{3} (x - 3)^2 \right] \,dx = 6. \end{align*}

Now consider D as a Type II region, so D = \big\{(x,y)\,| \, 0 \leq y \leq 2, \space 0 \leq x \leq 3 - \frac{3}{2}y \big\}. In this calculation, the volume is

\begin{align*} V &= \int_{y=0}^{y=2} \int_{x=0}^{x=3-(3y/2)} (6 - 2x - 3y)\,dx \space dy = \int_{y=0}^{y=2} \left[(6x - x^2 - 3xy)\Big|_{x=0}^{x=3-(3y/2)} \right] \,dy \\[4pt] &= \int_{y=0}^{y=2} \left[\frac{9}{4}(y - 2)^2 \right] \,dy = 6.\end{align*}

Therefore, the volume is 6 cubic units.

Exercise \PageIndex{6}

Find the volume of the solid bounded above by f(x,y) = 10 - 2x + y over the region enclosed by the curves y = 0 and y = e^x where x is in the interval [0,1].

Hint

Sketch the region, and describe it as Type I.

Answer

\frac{e^2}{4} + 10e - \frac{49}{4} cubic units

Finding the area of a rectangular region is easy, but finding the area of a non-rectangular region is not so easy. As we have seen, we can use double integrals to find a rectangular area. As a matter of fact, this comes in very handy for finding the area of a general non-rectangular region, as stated in the next definition.

Definition: Double Integrals

The area of a plane-bounded region D is defined as the double integral

\iint\limits_D 1\,dA. \nonumber

We have already seen how to find areas in terms of single integration. Here we are seeing another way of finding areas by using double integrals, which can be very useful, as we will see in the later sections of this chapter.

Example \PageIndex{8}: Finding the Area of a Region

Find the area of the region bounded below by the curve y = x^2 and above by the line y = 2x in the first quadrant (Figure \PageIndex{13}).

The line y = 2 x (also marked x = y/2) is shown, as is y = x squared (also marked x = the square root of y). There are vertical and horizontal shadings giving for small stretch of this region, denoting that it can be treated as a Type I or Type II area.
Figure \PageIndex{13}: The region bounded by y = x^2 and y = 2x.
Solution

We just have to integrate the constant function f(x,y) = 1 over the region. Thus, the area A of the bounded region is \displaystyle \int_{x=0}^{x=2} \int_{y=x^2}^{y=2x} dy \space dx \space \text{or} \space \int_{y=0}^{y=4} \int_{x=y/2}^{x=\sqrt{y}} dx \space dy:

\begin{align*} A &= \iint\limits_D 1\,dx \space dy \\[4pt] &= \int_{x=0}^{x=2} \int_{y=x^2}^{y=2x} 1\,dy \space dx \\[4pt] &= \int_{x=0}^{x=2} \left(y\Big|_{y=x^2}^{y=2x} \right) \,dx \\[4pt] &= \int_{x=0}^{x=2} (2x - x^2)\,dx \\[4pt] &= \left(x^2 - \frac{x^3}{3}\right) \Big|_0^2 = \frac{4}{3}. \end{align*}

Exercise \PageIndex{7}

Find the area of a region bounded above by the curve y = x^3 and below by y = 0 over the interval [0,3].

Hint

Sketch the region.

Answer

\frac{81}{4} square units

We can also use a double integral to find the average value of a function over a general region. The definition is a direct extension of the earlier formula.

Definition: The Average Value of a Function

If f (x,y) is integrable over a plane-bounded region D with positive area A(D), then the average value of the function is

f_{ave} = \frac{1}{A(D)} \iint\limits_D f(x,y) \,dA. \nonumber

Note that the area is \displaystyle A(D) = \iint\limits_D 1\,dA.

Example \PageIndex{9}: Finding an Average Value

Find the average value of the function f(x,y) = 7xy^2 on the region bounded by the line x = y and the curve x = \sqrt{y} (Figure \PageIndex{14}).

The lines x = y and x = the square root of y bound a shaded region. There are horizontal dashed lines marked throughout the region.
Figure \PageIndex{14}: The region bounded by x = y and x = \sqrt{y}.
Solution

First find the area A(D) where the region D is given by the figure. We have

A(D) = \iint\limits_D 1\,dA = \int_{y=0}^{y=1} \int_{x=y}^{x=\sqrt{y}} 1\,dx \space dy = \int_{y=0}^{y=1} \left[x \Big|_{x=y}^{x=\sqrt{y}} \right] \,dy = \int_{y=0}^{y=1} (\sqrt{y} - y) \,dy = \frac{2}{3}\left. y^{2/3} - \frac{y^2}{2} \right|_0^1 = \frac{1}{6} \nonumber

Then the average value of the given function over this region is

\begin{align*} f_{ave} = \frac{1}{A(D)} \iint\limits_D f(x,y) \,dA = \frac{1}{A(D)} \int_{y=0}^{y=1}\int_{x=y}^{x=\sqrt{y}} 7xy^2 \,dx \space dy = \frac{1}{1/6} \int_{y=0}^{y=1} \left[ \left. \frac{7}{2} x^2y^2 \right|_{x=y}^{x=\sqrt{y}} \right] \,dy \\ = 6 \int_{y=0}^{y=1} \left[ \frac{7}{2} y^2 (y - y^2)\right] \,dy = 6\int_{y=0}^{y=1} \left[ \frac{7}{2} (y^3 -y^4) \right] \,dy = \frac{42}{2} \left. \left( \frac{y^4}{4} - \frac{y^5}{5}\right) \right|_0^1 = \frac{42}{40} = \frac{21}{20}. \end{align*}

Exercise \PageIndex{8}

Find the average value of the function f(x,y) = xy over the triangle with vertices (0,0), \space (1,0) and (1,3).

Hint

Express the line joining (0,0) and (1,3) as a function y = g(x).

Answer

\frac{3}{4}

Improper Double Integrals

An improper double integral is an integral \displaystyle \iint\limits_D f \,dA where either D is an unbounded region or f is an unbounded function. For example, D = \big\{(x,y) \,|\,|x - y| \geq 2\big\} is an unbounded region, and the function f(x,y) = 1/(1 - x^2 - 2y^2) over the ellipse x^2 + 3y^2 \geq 1 is an unbounded function. Hence, both of the following integrals are improper integrals:

  1. \iint\limits_D xy \space dA \space \text{where} \space D = \big\{(x,y)| | \, x - y| \geq 2 \big\}; \nonumber
  2. \iint\limits_D \frac{1}{1 - x^2 -2y^2}\,dA \space \text{where} \space D = \big\{(x,y)| \, x^2 + 3y^2 \leq 1 \big\}. \nonumber

In this section we would like to deal with improper integrals of functions over rectangles or simple regions such that f has only finitely many discontinuities. Not all such improper integrals can be evaluated; however, a form of Fubini’s theorem does apply for some types of improper integrals.

Theorem: Fubini’s Theorem for Improper Integrals

If D is a bounded rectangle or simple region in the plane defined by

\big\{(x,y)\,: a \leq x \leq b, \space g(x) \leq y \leq h(x) \big\} and also by

\big\{(x,y)\,: c \leq y \leq d, \space j(y) \leq x \leq k(y)\big\} and f is a nonnegative function on D with finitely many discontinuities in the interior of D then

\iint\limits_D f \space dA = \int_{x=a}^{x=b} \int_{y=g(x)}^{y=h(x)} f(x,y) \,dy \space dx = \int_{y=c}^{y=d} \int_{x=j(y)}^{x=k(y)} f(x,y) \,dx \space dy \nonumber

It is very important to note that we required that the function be nonnegative on D for the theorem to work. We consider only the case where the function has finitely many discontinuities inside D.

Example \PageIndex{10}: Evaluating a Double Improper Integral

Consider the function f(x,y) = \frac{e^y}{y} over the region D = \big\{(x,y)\,: 0 \leq x \leq 1, \space x \leq y \leq \sqrt{x}\big\}.

Notice that the function is nonnegative and continuous at all points on D except (0,0). Use Fubini’s theorem to evaluate the improper integral.

Solution

First we plot the region D (Figure \PageIndex{15}); then we express it in another way.

The line y = x is shown, as is y = the square root of x.
Figure \PageIndex{15}: The function f is continuous at all points of the region D except (0,0).

The other way to express the same region D is

D = \big\{(x,y)\,: \, 0 \leq y \leq 1, \space y^2 \leq x \leq y \big\}. \nonumber

Thus we can use Fubini’s theorem for improper integrals and evaluate the integral as

\int_{y=0}^{y=1} \int_{x=y^2}^{x=y} \frac{e^y}{y} \,dx \space dy. \nonumber

Therefore, we have

\int_{y=0}^{y=1} \int_{x=y^2}^{x=y} \frac{e^y}{y} \,dx \space dy = \int_{y=0}^{y=1} \left. \frac{e^y}{y}x\right|_{x=y^2}^{x=y} \,dy = \int_{y=0}^{y=1} \frac{e^y}{y} (y - y^2) \,dy = \int_0^1 (e^y - ye^y)\,dy = e - 2. \nonumber

As mentioned before, we also have an improper integral if the region of integration is unbounded. Suppose now that the function f is continuous in an unbounded rectangle R.

Theorem: Improper Integrals on an Unbounded Region

If R is an unbounded rectangle such as R = \big\{(x,y)\,: \, a \leq x \leq \infty, \space c \leq y \leq \infty \big\}, then when the limit exists, we have

\iint\limits_R f(x,y) \,dA = \lim_{(b,d) \rightarrow (\infty, \infty)} \int_a^b \left(\int_c^d f (x,y) \,dy \right) dx = \lim_{(b,d) \rightarrow (\infty, \infty)} \int_c^d \left(\int_a^b f(x,y) \,dx \right) dy. \nonumber

The following example shows how this theorem can be used in certain cases of improper integrals.

Example \PageIndex{11}

Evaluate the integral \iint\limits_R xye^{-x^2-y^2}\,dA where R is the first quadrant of the plane.

Solution

The region R is the first quadrant of the plane, which is unbounded. So

\begin{align*} \iint\limits_R xye^{-x^2-y^2} \,dA &= \lim_{(b,d) \rightarrow (\infty, \infty)} \int_{x=0}^{x=b} \left(\int_{y=0}^{y=d} xye^{-x^2-y^2} dy\right) \,dx \\ &= \lim_{(b,d) \rightarrow (\infty, \infty)} \int_{y=0}^{x=b} xye^{-x^2-y^2} \,dy \\ &= \lim_{(b,d) \rightarrow (\infty, \infty)} \frac{1}{4} \left(1 - e^{-b^2}\right) \left( 1 - e^{-d^2}\right) = \frac{1}{4} \end{align*}

Thus,

\iint\limits_R xye^{-x^2-y^2}\,dA \nonumber

is convergent and the value is \frac{1}{4}.

Exercise \PageIndex{9}

\iint\limits_D \frac{y}{\sqrt{1 - x^2 - y^2}}dA \nonumber where D = \big\{(x,y)\,: \, x \geq 0, \space y \geq 0, \space x^2 + y^2 \leq 1 \big\}.

Hint

Notice that the integral is nonnegative and discontinuous on x^2 + y^2 = 1. Express the region D as D = \big\{(x,y)\,: \, 0 \leq x \leq 1, \space 0 \leq y \leq \sqrt{1 - x^2} \big\} and integrate using the method of substitution.

Answer

\frac{\pi}{4}

In some situations in probability theory, we can gain insight into a problem when we are able to use double integrals over general regions. Before we go over an example with a double integral, we need to set a few definitions and become familiar with some important properties.

Definition: Joint Density Function

Consider a pair of continuous random variables X and Y such as the birthdays of two people or the number of sunny and rainy days in a month. The joint density function f of X and Y satisfies the probability that (X,Y) lies in a certain region D:

P((X,Y) \in D) = \iint\limits_D f(x,y) \,dA. \nonumber

Since the probabilities can never be negative and must lie between 0 and 1 the joint density function satisfies the following inequality and equation:

f(x,y) \geq 0 \space \text{and} \space \iint\limits_R f(x,y) \,dA = 1. \nonumber

Definition: Independent Random Variables

The variables X and Y are said to be independent random variables if their joint density function is the product of their individual density functions:

f(x,y) = f_1(x) f_2(y). \nonumber

Example \PageIndex{12}: Application to Probability

At Sydney’s Restaurant, customers must wait an average of 15 minutes for a table. From the time they are seated until they have finished their meal requires an additional 40 minutes, on average. What is the probability that a customer spends less than an hour and a half at the diner, assuming that waiting for a table and completing the meal are independent events?

Solution

Waiting times are mathematically modeled by exponential density functions, with m being the average waiting time, as

f(t) = \begin{cases} 0, & \text{if}\; t<0 \\ \dfrac{1}{m}e^{-t/m}, & \text{if} \; t\geq 0.\end{cases} \nonumber

if X and Y are random variables for ‘waiting for a table’ and ‘completing the meal,’ then the probability density functions are, respectively,

f_1(x) = \begin{cases} 0, & \text{if}\; x<0. \\ \dfrac{1}{15} e^{-x/15}, & \text{if} \; x\geq 0. \end{cases} \quad \text{and} \quad f_2(y) = \begin{cases} 0, & \text{if}\; y<0 \\ \dfrac{1}{40} e^{-y/40}, & \text{if}\; y\geq 0. \end{cases} \nonumber

Clearly, the events are independent and hence the joint density function is the product of the individual functions

f(x,y) = f_1(x)f_2(y) = \begin{cases} 0, & \text{if} \; x<0 \; \text{or} \; y<0, \\ \dfrac{1}{600} e^{-x/15}, & \text{if} \; x,y\geq 0 \end{cases} \nonumber

We want to find the probability that the combined time X + Y is less than 90 minutes. In terms of geometry, it means that the region D is in the first quadrant bounded by the line x + y = 90 (Figure \PageIndex{16}).

The line x + y = 90 is shown.
Figure \PageIndex{16}: The region of integration for a joint probability density function.

Hence, the probability that (X,Y) is in the region D is

P(X + Y \leq 90) = P((X,Y) \in D) = \iint\limits_D f(x,y) \,dA = \iint\limits_D \frac{1}{600}e^{-x/15} e^{-y/40} \,dA. \nonumber

Since x + y = 90 is the same as y = 90 - x, we have a region of Type I, so

\begin{align*} D &= \big\{(x,y)\,|\,0 \leq x \leq 90, \space 0 \leq y \leq 90 - x\big\}, \\[6pt] P(X + Y \leq 90) &= \frac{1}{600} \int_{x=0}^{x=90} \int_{y=0}^{y=90-x} e^{-(/15}e^{-y/40}dx \space dy = \frac{1}{600} \int_{x=0}^{x=90} \int_{y=0}^{y=90-x}e^{-x/15}e^{-y/40} dx \space dy \\[6pt] &= \frac{1}{600} \int_{x=0}^{x=90} \int_{y=0}^{y=90-x} e^{-(x/15+y/40)}dx \space dy = 0.8328 \end{align*}

Thus, there is an 83.2\% chance that a customer spends less than an hour and a half at the restaurant.

Another important application in probability that can involve improper double integrals is the calculation of expected values. First we define this concept and then show an example of a calculation.

Definition: Expected Values

In probability theory, we denote the expected values E(X) and E(Y) respectively, as the most likely outcomes of the events. The expected values E(X) and E(Y) are given by

E(X) = \iint\limits_S x\,f(x,y) \,dA \space and \space E(Y) = \iint\limits_S y\,f (x,y) \,dA, \nonumber

where S is the sample space of the random variables X and Y.

Example \PageIndex{13}: Finding Expected Value

Find the expected time for the events ‘waiting for a table’ and ‘completing the meal’ in Example \PageIndex{12}.

Solution

Using the first quadrant of the rectangular coordinate plane as the sample space, we have improper integrals for E(X) and E(Y). The expected time for a table is

\begin{align*} E(X) &= \iint\limits_S x\frac{1}{600} e^{-x/15}e^{-y/40} \, dA\\[6pt] &= \frac{1}{600} \int_{x=0}^{x=\infty} \int_{y=0}^{y=\infty} xe^{-x/15} e^{-y/40}dA\\[6pt] &= \frac{1}{600} \lim_{(a,b) \rightarrow (\infty, \infty)} \int_{x=0}^{x=a} \int_{y=0}^{y=b} xe^{-x/15} e^{-y/40} dx \space dy \\[6pt] &= \frac{1}{600} \left(\lim_{a\rightarrow \infty}\int_{x=0}^{x=a} xe^{-x/15} dx \right) \left( \lim_{b\rightarrow \infty} \int_{y=0}^{y=b} e^{-y/40} dy \right) \\[6pt] &= \frac{1}{600}\left(\left. (\lim_{a\rightarrow \infty} (-15e^{-x/15}(x + 15)))\right|_{x=0}^{x=a} \right) \left( \left. ( \lim_{b\rightarrow \infty}(-40e^{-y/40})) \right|_{y=0}^{y=b}\right) \\[6pt] &=\frac{1}{600} \left( \lim_{a \rightarrow \infty} (-15e^{-a/15} (x + 15) + 225) \right)\left(\lim_{b\rightarrow \infty} (-40e^{-b/40} + 40) \right) \\[6pt] &= \frac{1}{600} (225)(40) = 15. \end{align*}

A similar calculation shows that E(Y) = 40. This means that the expected values of the two random events are the average waiting time and the average dining time, respectively.

Exercise \PageIndex{10}

The joint density function for two random variables X and Y is given by

f(x,y) =\begin{cases}\frac{1}{600} (x^2 + y^2),\; & \text{if} \; \leq x \leq 15, \; 0 \leq y \leq 10 \\ 0, & \text{otherwise} \end{cases} \nonumber

Find the probability that X is at most 10 and Y is at least 5.

Hint

Compute the probability

P(X \leq 10, \space Y \geq 5) = \int_{x=-\infty}^{10} \int_{y=5}^{y=10} \frac{1}{6000} (x^2 + y^2) dy \space dx. \nonumber

Answer

\frac{55}{72} \approx 0.7638

Key Concepts

  • A general bounded region D on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.
  • To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.
  • We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.
  • We can use Fubini’s theorem for improper integrals to evaluate some types of improper integrals.

Key Equations

  • Iterated integral over a Type I region

\iint\limits_D f(x,y) \,dA = \iint\limits_D f(x,y) \,dy \space dx = \int_a^b \left[\int_{g_1(x)}^{g_2(x)} f(x,y) \,dy \right] dx \nonumber

  • Iterated integral over a Type II region

\iint\limits_D f(x,y) \,dA = \iint\limits_D (x,y) \,dx \space dy = \int_c^d \left[ \int_{h_1(y)}^{h_2(y)} f(x,y) \,dx \right] dy \nonumber

Glossary

improper double integral
a double integral over an unbounded region or of an unbounded function
Type I
a region D in the xy- plane is Type I if it lies between two vertical lines and the graphs of two continuous functions g_1(x) and g_2(x)
Type II
a region D in the xy-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions h_1(y) and h_2(h)

This page titled 3.3: Double Integrals Over General Regions is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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