BMTC-131
CALCULUS
Indira Gandhi National Open University
School of Sciences
Block
5
INTEGRATION
Block Introduction 3
Notations and Symbols 4
UNIT 17
Introduction to Integration 5
UNIT 18
Methods of Integration 37
UNIT 19
Reduction Formulas 98
UNIT 20
Applications of Integration 126
Miscellaneous Examples and Exercises 161
,Course Design Committee*
Prof. Rashmi Bhardwaj Prof. Meena Sahai
G.G.S. Indraprastha University, Delhi University of Lucknow
Dr. Sunita Gupta Dr. Sachi Srivastava
University of Delhi University of Delhi
Prof. Amber Habib Faculty Members
Shiv Nadar University School of Sciences, IGNOU
Gautam Buddha Nagar Prof. M. S. Nathawat (Director)
Prof. S. A. Katre Dr. Deepika
University of Pune Mr. Pawan Kumar
Prof. V. Krishna Kumar Prof. Poornima Mital
NISER, Bhubaneswar Prof. Parvin Sinclair
Prof. Sujatha Varma
Dr. Amit Kulshreshtha
Dr. S. Venkataraman
IISER, Mohali
Dr. Aparna Mehra
I.I.T. Delhi
Prof. Rahul Roy
Indian Statistical Institute, Delhi
* The course design is based on the recommendations of the Programme Expert
Committee and the UGC- CBCS template
Block Preparation Team
Prof. Amber Habib (Editor) Dr. S. Venkataraman
Shiv Nadar University School of Sciences
Gautam Buddha Nagar IGNOU
Course Coordinators: Prof. Parvin Sinclair and Dr. Deepika
Acknowledgement: To Prof. Parvin Sinclair for comments on the manuscript. Parts of
this block are based on the previous course Calculus (MTE-01).
July, 2019
©Indira Gandhi National Open University, 2019
ISBN:
All rights reserved. No part of this work may be reproduced in any form, by mimeograph or any other
means, without written permission from the Indira Gandhi National Open University.
Further information on the Indira Gandhi National Open University courses may be obtained from the
University's of ce at Maidan Garhi, New Delhi-110 068 or from the University's website
http://www.ingou.ac.in.
Printed and Published on behalf of Indira Gandhi National Open University, New Delhi, by Prof. M. S.
Nathawat, Director, School of Sciences.
,BLOCK 5 INTEGRATION
The ancients knew how to nd the area and circumference of a circle. They also knew
how to nd the area of many other regular gures like polygons. However, they were not
able to nd the areas, lengths and volumes of irregular gures with the methods
available to them. These problems were solved by the Europeans only after
renaissance. The important concept of in nitesimally small objects and how to add them
up was found only in this period. Many of the insights were gained from the study of
problems of Astronomy, Geometry and physical sciences. In the seventeenth century
with the insights of their predecessors like Isaac Barrow and others, Newton and Leibniz
laid the foundations of Calculus. Their work was improved upon by their successors like
the Bernoulli family, Euler, Lagrange, Laplace and later on by Cauchy, Riemann and
others.
In the second, third and fourth blocks of this course, we discussed the process of
differentiation. We also saw how differentiation is a useful tool in Mathematics. In this
block, we will study integration which can be viewed as the reverse process of
differentiation.
In Unit 17, we introduce you to integration. In this unit we will see how the area below
the graph of a function can be approximated by sums of areas of rectangles. As the
number of rectangles increase and their sizes grow smaller and smaller their sum
approaches the area. We will also see the Fundamental Theorem of Calculus which
links the process of integration and differentiation. We will also see how to integrate
some simple functions here.
In Unit 18, we focus on techniques for integrating a variety of functions like rational
functions, trigonometric and functions that involve square root of a degree two
polynomial. The back bone of all these methods is the method of substitution that we
discuss at the beginning of this unit. We will also see some other techniques like
integration by parts. We also see how to use partial fractions to integrate rational
functions.
In Unit 19, we will discuss reduction formulas. The reduction formulas help us in
integrating some common functions that we come across in applications. The main idea
behind a reduction formula is to progressively reduce a parameter, usually the power of
x or a trigonometric function in the integrand, using integration by parts. This results in a
simpler integrand which we can integrate using standard methods.
In Unit 20, we will see how we can nd the areas bounded by plane curves and the
length of plane curves by a variety of methods. We begin by discussing the method for
nding the area between the graphs of two functions when the graph is given in
cartesian coordinates. We then discuss how to nd the areas of curves in plane given in
polar and parametric form. We will also discuss how to nd the length of the curves
given in cartesian, polar and parametric forms.
Finally, at the end of this Block, we have given some miscellaneous examples and
exercises. We hope that they will be useful in reinforcing your understanding of the
material in the units.
A word about some signs used in the unit! Throughout each unit, you will nd theorems,
examples and exercises. To signify the end of the proof of a theorem, we use the sign
. To show the end of an example, we use ∗ ∗ ∗. Further, equations that need to be
referred to are numbered sequentially within a unit, as are exercises and gures. E1, E2
etc. denote the exercises and Fig. 1, Fig. 2, etc. denote the gures.
3
, NOTATIONS AND SYMBOLS
L(P, f) Lower Sum, page 10
U(P, f) Upper Sum, page 10
P([a, b]) Set of partitions on [a, b], page 8
Rb
a f(x)dx Upper integral of the function f(x) over the interval [a, b]., page 15
f(x) dx Inde nite Integral of f(x), page 25
R
Rb
a f(x) dx Lower integral of the function over the interval [a, b]., page 15
L(f) {L(P, f)|P is a partition of [a, b]}, page 15
U(f) {U(P, f)|P is a partition of [a, b]}, page 15
For other notations and symbols, please refer to the lists in the previous blocks.
4
CALCULUS
Indira Gandhi National Open University
School of Sciences
Block
5
INTEGRATION
Block Introduction 3
Notations and Symbols 4
UNIT 17
Introduction to Integration 5
UNIT 18
Methods of Integration 37
UNIT 19
Reduction Formulas 98
UNIT 20
Applications of Integration 126
Miscellaneous Examples and Exercises 161
,Course Design Committee*
Prof. Rashmi Bhardwaj Prof. Meena Sahai
G.G.S. Indraprastha University, Delhi University of Lucknow
Dr. Sunita Gupta Dr. Sachi Srivastava
University of Delhi University of Delhi
Prof. Amber Habib Faculty Members
Shiv Nadar University School of Sciences, IGNOU
Gautam Buddha Nagar Prof. M. S. Nathawat (Director)
Prof. S. A. Katre Dr. Deepika
University of Pune Mr. Pawan Kumar
Prof. V. Krishna Kumar Prof. Poornima Mital
NISER, Bhubaneswar Prof. Parvin Sinclair
Prof. Sujatha Varma
Dr. Amit Kulshreshtha
Dr. S. Venkataraman
IISER, Mohali
Dr. Aparna Mehra
I.I.T. Delhi
Prof. Rahul Roy
Indian Statistical Institute, Delhi
* The course design is based on the recommendations of the Programme Expert
Committee and the UGC- CBCS template
Block Preparation Team
Prof. Amber Habib (Editor) Dr. S. Venkataraman
Shiv Nadar University School of Sciences
Gautam Buddha Nagar IGNOU
Course Coordinators: Prof. Parvin Sinclair and Dr. Deepika
Acknowledgement: To Prof. Parvin Sinclair for comments on the manuscript. Parts of
this block are based on the previous course Calculus (MTE-01).
July, 2019
©Indira Gandhi National Open University, 2019
ISBN:
All rights reserved. No part of this work may be reproduced in any form, by mimeograph or any other
means, without written permission from the Indira Gandhi National Open University.
Further information on the Indira Gandhi National Open University courses may be obtained from the
University's of ce at Maidan Garhi, New Delhi-110 068 or from the University's website
http://www.ingou.ac.in.
Printed and Published on behalf of Indira Gandhi National Open University, New Delhi, by Prof. M. S.
Nathawat, Director, School of Sciences.
,BLOCK 5 INTEGRATION
The ancients knew how to nd the area and circumference of a circle. They also knew
how to nd the area of many other regular gures like polygons. However, they were not
able to nd the areas, lengths and volumes of irregular gures with the methods
available to them. These problems were solved by the Europeans only after
renaissance. The important concept of in nitesimally small objects and how to add them
up was found only in this period. Many of the insights were gained from the study of
problems of Astronomy, Geometry and physical sciences. In the seventeenth century
with the insights of their predecessors like Isaac Barrow and others, Newton and Leibniz
laid the foundations of Calculus. Their work was improved upon by their successors like
the Bernoulli family, Euler, Lagrange, Laplace and later on by Cauchy, Riemann and
others.
In the second, third and fourth blocks of this course, we discussed the process of
differentiation. We also saw how differentiation is a useful tool in Mathematics. In this
block, we will study integration which can be viewed as the reverse process of
differentiation.
In Unit 17, we introduce you to integration. In this unit we will see how the area below
the graph of a function can be approximated by sums of areas of rectangles. As the
number of rectangles increase and their sizes grow smaller and smaller their sum
approaches the area. We will also see the Fundamental Theorem of Calculus which
links the process of integration and differentiation. We will also see how to integrate
some simple functions here.
In Unit 18, we focus on techniques for integrating a variety of functions like rational
functions, trigonometric and functions that involve square root of a degree two
polynomial. The back bone of all these methods is the method of substitution that we
discuss at the beginning of this unit. We will also see some other techniques like
integration by parts. We also see how to use partial fractions to integrate rational
functions.
In Unit 19, we will discuss reduction formulas. The reduction formulas help us in
integrating some common functions that we come across in applications. The main idea
behind a reduction formula is to progressively reduce a parameter, usually the power of
x or a trigonometric function in the integrand, using integration by parts. This results in a
simpler integrand which we can integrate using standard methods.
In Unit 20, we will see how we can nd the areas bounded by plane curves and the
length of plane curves by a variety of methods. We begin by discussing the method for
nding the area between the graphs of two functions when the graph is given in
cartesian coordinates. We then discuss how to nd the areas of curves in plane given in
polar and parametric form. We will also discuss how to nd the length of the curves
given in cartesian, polar and parametric forms.
Finally, at the end of this Block, we have given some miscellaneous examples and
exercises. We hope that they will be useful in reinforcing your understanding of the
material in the units.
A word about some signs used in the unit! Throughout each unit, you will nd theorems,
examples and exercises. To signify the end of the proof of a theorem, we use the sign
. To show the end of an example, we use ∗ ∗ ∗. Further, equations that need to be
referred to are numbered sequentially within a unit, as are exercises and gures. E1, E2
etc. denote the exercises and Fig. 1, Fig. 2, etc. denote the gures.
3
, NOTATIONS AND SYMBOLS
L(P, f) Lower Sum, page 10
U(P, f) Upper Sum, page 10
P([a, b]) Set of partitions on [a, b], page 8
Rb
a f(x)dx Upper integral of the function f(x) over the interval [a, b]., page 15
f(x) dx Inde nite Integral of f(x), page 25
R
Rb
a f(x) dx Lower integral of the function over the interval [a, b]., page 15
L(f) {L(P, f)|P is a partition of [a, b]}, page 15
U(f) {U(P, f)|P is a partition of [a, b]}, page 15
For other notations and symbols, please refer to the lists in the previous blocks.
4
