Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.81

Definite Integral

In the previous part ( )

b

a

f x dx has been defined as a limit of a sum. There is an important theorem in

Integral Calculus known as Fundamental theorem of Integral Calculus which states :

If there exists a function φ (x) such that (^) dxdφ( )x f x= ( ) for every x in a ≤ x ≤ b and if ( )
b
a
f x dx exists, then
( ) ( ) ( )
b
a
f x dx b a= φ − φ.

( )

b

a

f x dx, is read as ‘Integral from a to b of f(x) dx^1 where a is lower limit and b is the upper limit.

Symbol : φ(b) (a)− φ is written as ( )

b

a

(^) φx (^) , which is read as φ (x) from a to b.

∴ ( ) ( ) ( ) ( )

b b

a a

f x dx= φ^ x^ = φb a .− φ

Reason for the name ‘definite integral’.

Let us take f(x) dx = φ (x) + c instead of ∫ f(x) dx = φ (x).

Now ( ) ( ) ( ) ( ) ( )

b b

a a

f x dx= φ x c+ = φ^ b c+ − φ^ a c b a .+^ = φ − φ

Here the arbitrary constant ‘c’ is absent and ( ) ( ) ( )

b

a

f x dx b a= φ − φ and hence it is known as definite integral

or in other words definite integral is unique.

Rule to Evaluate : ( )

b

a

f x dx. (i) Find the value of f(x) dx, leaving the arbitary constant.

(ii) In the value obtained, put x = b (upper limit) and x = a (lower limit).

(iii) Deduct the second value from the first value (after putting the values of x).

(iv) The result thus obtained will be the required value of the definite integral.

A Few Results :

b b

a a

kf(x)dx k f(x)dx,= k is constant