Let f(x) be a function defined in a closed interval [a , b].

Then the definite integral _{a}∫^{b}f(x)dx represents the algebraic sum of the areas of the region bounded by the curve y = f(x) and the x-axis between the lines x = a and x = b

All the regions lying above the x-axis have ‘ positive ‘ areas whereas those lying below the x-axis have ‘ negative ‘ areas.

__Trapizodial Rule__

For finding the definite integral of a linear portion of a curve we can use the formula of area of trapizium and the rule applied is called trapizodial rule.

In the adjoining figure AD represent a linear curve and x = a and x = b are the limits of definite integral than

_{a}∫^{b}f(x)dx = Area ABCD

= 1/2)(sum of parallel sides) × height

= (1/2)[f(a) + f(b)].(b – a)

**Example :** Evaluate:

**Solution:**

Here, f(x) = 2x + 3

f(6) = 15 and f(2) = 7

= 2 × 22 = 44.

__Definite integral as Limit Of a Sum :__

An alternative way of describing _{a}∫^{b}f(x)dx is that the definite integral _{a}∫^{b}f(x)dx is a limiting case of the summation of an infinite series, provided f(x) is continuous on [a, b]

The converse is also true i.e., if we have an infinite series of the above form, it can be expressed as a definite integral.

The method to evaluate the integral, as limit of the sum of an infinite series is known as Integration by First Principle.

**Example :** Evaluate :

Solution: Let L =

= 2 ln 5 – 4 + 2 tan^{-1} 2

L = 25 e^{(2tan-12 – 4)}

**Exercise 1: Evaluate :**

(i)

(ii)

(iii)

(iv)