Let f(x) be a function defined in a closed interval [a , b].
Then the definite integral a∫bf(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.
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∫bf(x)dx = Area ABCD
= 1/2)(sum of parallel sides) × height
= (1/2)[f(a) + f(b)].(b – a)
Example : Evaluate:
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∫bf(x)dx is that the definite integral a∫bf(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 :