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DEFINITE INTEGRATION & APPLICATIONS

INTRODUCTION

Let f(x) be a function defined on the interval [a, b] and F(x) be its antiderivative. Then
The above is called the second Fundamental Theorem of calculus.
is defined as the definite integral of f(x) from x = a to x = b.
The numbers a and b are called limits of integration. We write

EVALUATION OF DEFINITE INTEGRALS BY SUBSTITUTION

Consider a definite integral of the following form
To evaluate this integral we proceed as following
Step 1 : Substitute

Step 2 : Find the limits of integration in new system of variable, i.e. the lower limit is g(a) and the upper limit is g(b), and the integral is now

Step 3 :  Evaluate the integral so obtained by usual method.

FUNDAMENTAL PROPERTIES OF DEFINITE INTEGRALS

PROPERTY 1

The definite integral  is a unique number.
[Note that the indefinite integral  is not unique]

PROPERTY 2

The definite integral is independent of the change of variable. thus,  etc all denote the same quantity.

PROPERTY 3

If the limits of definite integral are interchanged then its value changes sign, that is

PROPERTY 4

If   then

GENERALIZATION
If a < c1 < c2 < ........... < cn–1 < cn < b then
The property 4 is very useful in solving the integrals of functions defined in steps or modulus function or greatest integer functions.

PROPERTY 5

PROPERTY 6

This is very important property, because using this property many definite integrals can be evaluated without first finding the corresponding indefinite integrals which may be difficult or sometimes impossible to find.
DEDUCTION
Let I =
  
=   
Example :
 

PROPERTY 7

DEDUCTION
Let
Example :

PROPERTY 8


DEDUCTION
Example 1 :   Evaluate  

            
Example 2 : As cos (2π – x) = cosx
                                   = 0     As cos (π – x) = –cosx
In general

PROPERTY 9


DEDUCTION
  • If
  • If
Example 1 :   Evaluate
Here
      
Example 2 :   I =  for is an odd function

PROPERTY 10

If f(x) is periodic function with period T,
[i.e. f(x + T) = f(x)] Then
(i)
(ii)
(iii)
Example 1 :   
[since |cos x| is a periodic function with period π]

PROPERTY 11 - LEIBNITZ RULE FOR DIFFERENTIATION UNDER INTEGRAL SIGN

If φ (x) and  are defined over [a b] and differentiable for every x and f(t) is continuous, then
Example :  Find

GEOMETRICAL INTERPRETATION OF DEFINITE INTEGRALS

The definite integral is numerically equal to the area bounded by the curve y = f (x), the x-axis and the ordinates x = a, and x = b, as shown by  shaded region in the adjacent figure

AREA FUNCTION
Consider the definite integral
.
It represents the area of the shaded region shown in the adjacent figure.
The area of this shaded region is a function of x. Let us denote this function by A(x). We call this function A(x) as  area function A (x) = .
Note : The area function can be used to obtain indefinite integral of f(x) if we know the definite integral.

FIRST FUNDAMENTAL THEOREM OF CALCULUS

Let the area  function be defined by A (x) =  for all x ≥ a. Then A’(x)  = f (x)

DEFINITE INTEGRAL AS LIMIT OF A SUM

The definite integral   can also be expressed as a limiting case of summation of an infinite series, provided f(x) is continuous on [a  b]. We define
,
In particular if a = 0 and b = 1,
we get

WORKING RULE
If a given series is expressible in the form , then its limit as  can be expressed as a definite integral by writing x for , dx for  and integral sign . The lower and upper limits of integration are the values of   for the least and the greatest values of r respectively. In general

AREAS OF BOUNDED REGIONS

  1. Area bounded by curve y = f(x) the ordinates x = a, x = b and x–axis. is given by
 
  1. Area bounded by curve x = f(y), the abscissae y = c, y = d and y-axis is given by
       
  1. Area bounded by the curves  y = f(x) and y = g(x) is given by
      
where, a and b are the ordinates of points of intersection

SOME VERY IMPORTANT DEFINITE INTEGRALS

          
[When m and n both even]
[when either of m & n is odd]
These are called WALLI’S FORMULA
, n is even

, n is odd

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