# 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 (2Ï€ – x) = cosx

= 0 As cos (Ï€ – 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.e. f(x + T) = f(x)] Then

(i)

(ii)

(iii)

Example 1 :

[since |cos x| is a periodic function with period Ï€]

#### PROPERTY 11 - LEIBNITZ RULE FOR DIFFERENTIATION UNDER INTEGRAL SIGN

If Ï† (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

- Area bounded by curve y = f(x) the ordinates x = a, x = b and x–axis. is given by

- Area bounded by curve x = f(y), the abscissae y = c, y = d and y-axis is given by

- 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