Advanced Caluclus Unit XI
With FAQ
26.12.12 (new)
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Advanced Caluclus Unit XI
integration of functions of two or more independent variables - Part i
Objectives
From this session a learner is expected to achieve the following
- Familiarize with integration of functions of two independent variables.
- Learn that representing a region in a plane by means of inequalities helps in the evalaution of double integrals.
- Study Fubini’s theorems.
- Understand the concepts through examples.
Contents
1. Introduction
2. Integration of functions of two variables - Double integration
3. Properties of Double Integrals
4. Evaluation of Double Integrals
5. Representation of Region of Integration by means of inequalities
6. Fubini’s Theorem (First form)
7. Fubini’s Theorem (Stronger form)
1. Introduction
Integration of functions of two or more independent variables can be handled similar to the problems of integrating functions of one variable (Fig. 1.) In this unit we discuss the integration of functions of two independent variables. and some examples that illustrates the way of finding system of inequalities that represents a given region in a plane. Fubini’s Theorem (First form) and Fubini’s Theorem (Stronger form) along with examples will be discussed.
2. Integration of functions of two variables - Double integration
We first recall that if is a function of two independent variables, then the domain of is a subset of the plane. i.e., the domain of a function of two
variables is a region in the plane.
Let be a continuous function of two variables and defined on a region R bounded by a closed curve C . Let the region R be divided, in an arbitrary manner, into n sub-regions. So as not to inroduce new symbols we will denote by both the subregions and their areas. In each subregion take a point (Fig. 2); we will then have n points:
.Now consider the sum …(1)
The above is anintegral sum of the function in the region R.
If in R, then each term may be represented geometrically as the volume of a small cylinder with base and altitude
The sum is the sum of the volumes of the indicated elementary cylinders, that is, the volume of a certain ‘step-like’ solid.
Consider an arbitary sequence of integral sums formed by means of the function for the given region R,
…(2)
for different ways of partitioning R into subregions . We shall assume that the maximum diameter of the subregions approaches zero as Then the following
theorem, which we give without proof, holds true.
Theorem 1 If a function is continuous in a closed region R, then the sequence (2) of integral sums (1) has a limit if the maximum diameter of the subregions approaches zero as This limit is the same for any sequence of type (2), that is, it is independent either of the way R is partitioned into subregions or of the choice of the point inside a subregion.
This limit is called thedouble integral of over the regionR, and is denoted by That is, (3)
The region R is called the region of integration (this region R corresponds to the interval of integration in the case of definite integral of the form of a function of single variable).
In order to simplify the evaluation of the double integral one common choice for sub regions Riis the rectangular sub regions or rectangular grids, obtained by subdividing the region R by lines parallel to the coordinate axes as in Fig.3. Since the area of a typical rectangular grid is it follows from (3) that …(4)
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Properties of Double Integrals
if on ,where and
Property 5 above is called domain additivity property.
Evaluation of double integrals
Suppose that the region R is described by the inequalities
c y d, and g(y) x h(y) ; then
…(5)
Integral on the right of Eq.(5) is called a two fold iterated integral of over R. The inner integral is first evaluated with respect to x. Since the limits of inner integral are functions of y, the inner integral would be a function of y , say (y). Hence (5) becomes
…(6)
which can be easily evaluated.
Alternatively if the region R is described by the inequalities
a x b and r(x) y s(x) , then
…(7)
The inner integralis a function of x,sayhence (7) becomes …(8)
which can be easily be evaluated.
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Representation of Region of Integration by means of inequalities
We recall that, a convenient and simple way of describing a region Rin the is by giving a system of inequalities. Also, equations (5) to (8) discussed above can be used only when the region is given in terms of inequalities. Before going to evaluate double integrals, we first illustrate the way of finding system of inequalities associated with a given region in a plane.
Example 1LetR be the region inside and on the boundary of the rectangle bounded by the lines x = a, x = b; y = c, y = d. Write the region as a system of inequalities.
Solution
First of all draw the rectangular axes and the lines and (Fig. 4).
Consider strips parallel to the (Fig.5).Then the points on the strip varies from the line y = c to y = d . Hence limits of y are y = c and y = d. To cover the entire region such strip should start with the minimum value of x (here on the line x =a)to the maximum value of x (here on the line x = b). Hence limits of x are x = a, x = b.
Hence, the given region can be expressed by means of the following system of inequalities:
a x b; c y d.
Example 2Express by means of inequalities, the region enclosed by the triangle in the XY plane bounded by the X axis and the lines y = x and x =1.
Solution
First of all draw the rectangular axes and the lines y = x and x = 1 and obtain the region enclosed by these lines (Fig.6). Consider strips parallel to the (Fig. 7). Then the points on the strip varies from the line y = 0 to y = x. Hence limits of y are y = 0 and y = x. To cover the entire region such strips should start with the minimum value of x (here on the line x = 0) to the maximum value of x (here on the line x = 1). Hence limits of x are x = 0 and x = 1.
Hence, the given region can be expressed by means of the following system of inequalities:
0 x 1, 0 y x. . . . (9)
Alternative form:Consider strips parallel to the(Fig. 8). Then the points on the strip varies from the line to . Hence limits of x are and . To cover the entire region such strips should start with the minimum value of (here on the line ) to the maximum value of (here on the line ). Hence limits of y are and .
Hence, the given region can be expressed by means of the following system of inequalities:
0 y 1, y x 1. . . . (10)
Remark: In the two alternative representations given by inequalities above, set (9) of inequalities, in which x is bounded by constants, is used when integration is first with respect to y and then with respect to x. Set (10) of inequalities, in which y is bounded by constants, is used when integration is first with respect to x and then with respect to y.
Example 3Consider the region R enclosed by the ellipse Represent the region by means of system of inequalities.
Solution
First of all draw the ellipse which meets the rectangular axes at the points and then shade the region (Fig.9). Consider strips parallel to the (Fig.10). Then thepoints on the strip varies from the curve to the curve . Hence limits of y are
and .
To cover the entire region such strips should start with the minimum value of (on the line ) to the maximum value of (on the line ). Hence limits of are and Hence, the given region can be expressed by means of the following system of inequalities;
Alternate form: To get the alternative form, consider strips parallel to the(Fig 11). Then the points on the strip varies from the curve to the curve . Hence limits of are
and .
To cover the entire region such strips should start with the minimum value of (on the line ) to the maximum value of (on the line ). Hence limits of are and .
Hence, the given region can be expressed by means of the following system of inequalities;
Example 4Express by means of inequalities, the region bounded by the lines
Solution First of all draw the rectangular axes and the lines or . (Fig.12). Consider strips parallel to the (Fig.13). Then the points on the strip varies from the line y = 0 to . Hence limits of y are y = 0 and . To cover the entire region such strips should start with the minimum value of x (the line x = 0) to the maximum value of x (on the line ). Hence limits of x are and
Hence, the given region can be expressed by means of the following system of inequalities:
Alternate form: To get the alternative form, consider strips parallel to the-axis (Fig.14). Then the points on the strip varies from the line to . Hence limits of are and . To cover the entire region such strips should start with the minimum value of y (on the line y = 1) to the maximum value of y (on the line). Hence, the given region can be expressed by means of the following system of inequalities:
Example 5 Express the region of integration for the integral
by means of system of inequalities. Also find the alternate form.
Solution
The region of integration is given by the system of inequalities :
0 x 2 ; x2 y 2 x.
To get the alternative form we draw the lines and the parabola and shade the region and proceed as in earlier examples. Then we will get the alternative form as .
Example 6Find the alternate form of the region given by the system of inequalities
Solution First of all draw the co ordinate axes and the line y = x and shade the region bounded by and (Fig. 15). Here the limits of are constants. Hence to get the alternative form we consider strips parallel to the -axis (Fig.16).Then the points on the strip varies from the line to . Hence the limits of are and . To cover the entire region such strips should start with the minimum value of (on the line = 0) to the maximum value of (on the line = ). Hence, the given region can be expressed by means of the following system of inequalities;
.
6. Fubini’s Theorem (First form)
We now describe Fubini’s Theorem for calculating double integrals over rectangular regions.
If f (x, y) is continuous on the rectangular region R: a x b, c y d , thenboth of the following definitions give the value of .
…(11)
…(12)
Remark In definition (11), the integration is first with respect to x and then with respect ot y; while in definition (12), the integration is first with respect to y and then with respect ot x.
Example7 Evaluate for , R : 0 x 2, 1 y 1. Verify that the change in the order of integration doesn’t effect the result.
Solution
Reversing the order of integration, we have
Hence the order of integration doesn’t effect the result.
Example 8 Evaluate I =
Solution
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7. Fubini’s Theorem (Stronger form)
We now describe Fubini’s Theorem for calculating double integrals over bounded non-rectangular regions.
Let is continuous on a (rectangular or non-rectangular) region R.
1. If R is defined by
with and continuous on , then
…(13)
2. If R is defined by
with and continuous on , then
…(14)
Remark In (13) above integration is first with respect to y while in (14) integration is first with respect to x.
Example9 Evaluate
Solution
.
Example 10 Evaluate xy dx dy over the first quadrant of the circle .
Solution
Here the circle meets thewhen and this implies . Hence in the first quadrant the circle meets the line Consider strips parallel to the . Each strip varies from to the curve . To cover the entire region each strip move from . Hence the region enclosed in the first quadrant of the circle is given by the set of inequalities:
Example11Evaluate over the region R in which x 0; y 0 and
Solution Referring to an earlier example, we have the region of integration is
.
Hence
Summary
In this session integration of functions of two independent variables have been discussed and seen that double integral is the limit of integral sums. Properties of double integrals and the method of evaluating double integrals have been discussed. A number of examples have been discussed that illustrates the way of finding system of inequalities that represents a given region in a plane. Fubini’s Theorem (First form) and Fubini’s Theorem (Stronger form) along with examples have been discussed.
Assignments:
- Show that
- Evaluate
- Evaluate over the region R in which each of and .
- Evaluate where R is the region bounded by the ellipse.
- Evaluate where R is the region of the circle , .
References
- Gorakh Prasad, Integral Calculus, Pothishala Pvt. Ltd., Allahabad.
- N. Piskunov, Differential and Integral Calculus, Peace Publsihers, Moscow.
- Shanti Narayan, A course of Mathematical Analysis, S. Chand and Company, New Delhi.
Quiz
1. The value of the double integral is ______
(a)
(b)
(c)
(d)
Ans. (a)
2. The value of the double integral is ______
(a)
(b)
(c)
(d)
Ans. (d)
3. The value of the double integral is ______
(a) 6
(b)
(c) 0
(d) 2
Ans.(b)
4. The alternate form of the region is ______
(a)
(b)
(c)
(d)
Ans. (c)
5. If and , then which of the following is true?
(a)
(b)
(c) provided
(d) none of the above.
Ans.(b)
FAQ
1. Whether integration of a function of two independent variables defined over a region in a plane is possible?
Answer. Yes, possible. In such cases integrals are called double integrals.
2. What is the procedure for the evaluation of a given double integral?
Answer: The region of integration of a double integral is a region in a plane. If that region is represented by means of system of inequalities we determine the double integral using Fubini’s theorem, treating it is as two fold iterated integral of the function over the region. If the region is not immediately given in terms of system of inequalities, we first determine it.
3. Is there any geometrical significance to the double integral?
Answer. Yes. Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three dimensional Cartesian plane where z = ƒ(x, y)) and the plane which contains its domain.
Glossary
Integral sum of a function in a region: Let be a continuous function of two variables and defined on a region R bounded by a closed curve C . Let the region R be divided, in an arbitrary manner, into n sub-regions . So as not to inroduce new symbols we will denote by both the subregions and their areas. In each subregion take a point (Fig. 2); we will then have n points:
.
The sum
is an integral sum of the function in the region R.
Double integral of a function in a region: If a function is continuous in a closed region R, then the sequence of integral sums has a limit if the maximum diameter of the subregions approaches zero as This limit is the same for any sequence of integral sums, that is, it is independent either of the way R is partitioned into subregions or of the choice of the point inside a subregion . This limit is called the double integral of over the region R, and is denoted by
Region of integration: The region R under consideration of the double integral is called the region of integration.
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