MODULE-4
Multivariable Calculus- I
(Multiple Integration)
Syllabus: Multiple integral: Double integral, Triple integral, Change of order of integration, Change of
variables, Applications: Area, Volume, Centre of Mass & Centre of Gravity (constant and variable
densities).
Course Outcomes
Relate the multiple integral tools for Calculating area, volume, centre of mass and centre of
gravity.
TABLE OF CONTENTS
S.No. Topic Page No.
4.1 Course Objectives of Module 4 1
4.2 Introduction 2
4.3 Multiple Integration 2
4.3.1 Double Integral 3
4.3.2 Triple Integral 12
4.3.3 Change of order of integration 15
4.3.4 Change of variables 19
4.4 Applications 30
4.4.1 Area 30
4.4.2 Volume 36
4.4.3 Centre of mass & Centre of gravity 41
( Constant and variable densities)
4.5 Related Links 46
1
,4.2 INTRODUCTION
Multiple integral is a natural extension of a definite integral to a function of two variables (Double integral) or
three variables (triple integral) or more variables.
4.2.1 APPLICATIONS
➢ Double and triple integrals are useful in finding Area.
➢ Double and triple integrals are useful in finding Volume.
➢ Double and triple integrals are useful in finding Mass.
➢ Double and triple integrals are useful in finding Centroid.
➢ In finding Average value of a function.
➢ In finding Distance, Velocity, Acceleration.
➢ Useful in calculating Kinetic energy and Improper Integrals.
➢ In finding Arc Length of a curve.
➢ The most important application of Multiple Integrals involves finding areas bounded by a curve and
coordinate axes and area between two curves.
➢ It includes finding solutions to various complicated problems of work and energy.
➢ In mechanics, the moment of inertia is calculated as the volume integral (triple integral) of
the density weighed with the square of the distance from the axis.
➢ Multiple integrals are used in many applications in physics. The gravitational potential associated with a mass
distribution given by a mass measure on three-dimensional Euclidean space R3 is calculated by triple
integration.
➢ In electromagnetism, Maxwell's equations can be written using multiple integrals to calculate the total
magnetic and electric fields.
➢ We can determine the probability of an event if we know the probability density function using double
integration.
4.3 MULTIPLE INTEGRAL
4.3.1 Double integral
A double integral is its counterpart in two dimensions. Let a single valued and bounded function f(x, y) of two
independent variables x, y defined in a closed region R.
Then double integral of f(x, y) over the region R is denoted by,
Also we can express as or
2
,4.3.1.2 Evaluation of Double Integral in Cartesian Coordinates
The method of evaluating the double integrals depends upon the nature of the curves bounding the region R. Let the region
be bounded by the curves .
(i) When are functions of y and are constants: If we have functional limits of x in terms of dependent
variable y [ and constant limits of variable y then we will first integrate with respect to variable x
in case of double integral, as follows:
=
(Here we have drawn the strip parallel to x axis, because
variable limits [ are provided.)
(ii) When are functions of x and are constants: If we have functional limits of y in terms of dependent
variable x [ and constant limits of variable x then we will first integrate with respect to variable y
in case of double integral, as follows:
=
3
, (Here we have drawn the strip parallel to y axis, because variable limits [ are provided.)
(iii) When are constants: If we have both the variables x and y with constant limits then we
can first integrate with respect to any variable x or y in case of double integral, as follows:
=
=
(Here we can draw the strip parallel to any of the axes, because both x and y are having constant limits.)
From case no. (i) and (ii) discussed above, we observe that integration is to be performed w.r.t. the variable
limits first and then w.r.t. the variable with constant limits.
4.3.1.3 Solved examples
Example1: Evaluate .
Solution: = (Here we have constant limits for both x and y variables,
so we may integrate w.r.t. any variable the first)
4
Multivariable Calculus- I
(Multiple Integration)
Syllabus: Multiple integral: Double integral, Triple integral, Change of order of integration, Change of
variables, Applications: Area, Volume, Centre of Mass & Centre of Gravity (constant and variable
densities).
Course Outcomes
Relate the multiple integral tools for Calculating area, volume, centre of mass and centre of
gravity.
TABLE OF CONTENTS
S.No. Topic Page No.
4.1 Course Objectives of Module 4 1
4.2 Introduction 2
4.3 Multiple Integration 2
4.3.1 Double Integral 3
4.3.2 Triple Integral 12
4.3.3 Change of order of integration 15
4.3.4 Change of variables 19
4.4 Applications 30
4.4.1 Area 30
4.4.2 Volume 36
4.4.3 Centre of mass & Centre of gravity 41
( Constant and variable densities)
4.5 Related Links 46
1
,4.2 INTRODUCTION
Multiple integral is a natural extension of a definite integral to a function of two variables (Double integral) or
three variables (triple integral) or more variables.
4.2.1 APPLICATIONS
➢ Double and triple integrals are useful in finding Area.
➢ Double and triple integrals are useful in finding Volume.
➢ Double and triple integrals are useful in finding Mass.
➢ Double and triple integrals are useful in finding Centroid.
➢ In finding Average value of a function.
➢ In finding Distance, Velocity, Acceleration.
➢ Useful in calculating Kinetic energy and Improper Integrals.
➢ In finding Arc Length of a curve.
➢ The most important application of Multiple Integrals involves finding areas bounded by a curve and
coordinate axes and area between two curves.
➢ It includes finding solutions to various complicated problems of work and energy.
➢ In mechanics, the moment of inertia is calculated as the volume integral (triple integral) of
the density weighed with the square of the distance from the axis.
➢ Multiple integrals are used in many applications in physics. The gravitational potential associated with a mass
distribution given by a mass measure on three-dimensional Euclidean space R3 is calculated by triple
integration.
➢ In electromagnetism, Maxwell's equations can be written using multiple integrals to calculate the total
magnetic and electric fields.
➢ We can determine the probability of an event if we know the probability density function using double
integration.
4.3 MULTIPLE INTEGRAL
4.3.1 Double integral
A double integral is its counterpart in two dimensions. Let a single valued and bounded function f(x, y) of two
independent variables x, y defined in a closed region R.
Then double integral of f(x, y) over the region R is denoted by,
Also we can express as or
2
,4.3.1.2 Evaluation of Double Integral in Cartesian Coordinates
The method of evaluating the double integrals depends upon the nature of the curves bounding the region R. Let the region
be bounded by the curves .
(i) When are functions of y and are constants: If we have functional limits of x in terms of dependent
variable y [ and constant limits of variable y then we will first integrate with respect to variable x
in case of double integral, as follows:
=
(Here we have drawn the strip parallel to x axis, because
variable limits [ are provided.)
(ii) When are functions of x and are constants: If we have functional limits of y in terms of dependent
variable x [ and constant limits of variable x then we will first integrate with respect to variable y
in case of double integral, as follows:
=
3
, (Here we have drawn the strip parallel to y axis, because variable limits [ are provided.)
(iii) When are constants: If we have both the variables x and y with constant limits then we
can first integrate with respect to any variable x or y in case of double integral, as follows:
=
=
(Here we can draw the strip parallel to any of the axes, because both x and y are having constant limits.)
From case no. (i) and (ii) discussed above, we observe that integration is to be performed w.r.t. the variable
limits first and then w.r.t. the variable with constant limits.
4.3.1.3 Solved examples
Example1: Evaluate .
Solution: = (Here we have constant limits for both x and y variables,
so we may integrate w.r.t. any variable the first)
4
