Definite Integral

Definite Integral 

1. We know that the indefinite integral of a function f(x) is not unique and it varies by a constant, known as the constant of integration i.e. I = ... (1) where F(x) is the

anti-derivative of f(x) and c is the constant of integration.


2. In geometrical and other applications of integration, it becomes necessary to find the difference in the values of (1) for two different assigned values of the independent variable x, say

a and b. This difference is called the definite integral of f(x) over the interval (a, b) and is denoted by . Thus ..(2) a is called the lower and b, the upper limit of integration; the value of is unique.



Geometrical Interpretation of the Definite Integral

The represents the algebraic sum of the areas of the regions bounded by the graph of the function y = f(x), the x-axis and the straight lines x = a and x = b. The areas above the x-axis enter into this sum with plus sign, while those below the x-axis enter it with a minus sign

.


The definite integral can be evaluated by using various techniques of integration studied in indefinite integrals or by using the properties of definite integrals or by writing it as a limit of a sum.


Definite Integral as the Limit of a Sum

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


Note:

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

Method to express the infinite series as definite integral:

Express the given series in the form .

Then the limit is its sum when, i.e. .

Replace by x and by dx and by the sign of .

The lower and the upper limit of integration are the limiting values of for the first and the last term of r respectively.



Some particular cases of the above are

(a) or

(b) where (as r = 1) and (as r = pn)



Fundamental Theorem of Integral Calculus:

Let F(x) be an anti-derivative of a continuous function f(x) on [a, b]. Then.



Properties of Definite Integral

1. .

2. Change of variable of integration is immaterial so long as limits of integration remain the same i.e. .

3. .

4.

where the point c may lie between a and b or it may be exterior to (a, b).

Note:

(i) This property is useful when f (x) is not continuous in [a, b], because we can break up the integral into several integrals at the points of discontinuity so that the function is continuous in the subintervals.

(ii) This property is true even when c lies outside the interval [a, b].


5.


6.



Special cases:
If f (x) = f(a - x), then and If f (x) = - f(a - x), then .

7.


Special case:




8. If f (x) is a periodic function with period T, then


(a) , where n I.

(i) In particular, if a = 0

.

(ii) If n = 1

.

(b) , where m, n I.

(c) , where n I.




DIFFERENTIATION UNDER THE INTEGRAL SIGN

A. Leibnitz’s Rule

If g is continuous on [a, b] and f₁ (x) and f₂ (x) are differentiable functions whose values lie in

[a, b], then .

In particular .

B. If F(t) =, where represents the derivative of g with respect to t keeping x constant.



Inequalities

Sometimes you are asked to prove inequalities involving definite integrals or to estimate the upper and lower bounds of a definite integral, where the exact value of the definite integral is difficult to find. Under these circumstances, we use the following results:

(i) Equality sign holds when f (x) is entirely of the same sign on [a, b].

(ii) If f(x) g(x) on [a, b], then. In particular, if f(x) 0, then .


(iii) For a given function f (x) continuous on [a, b] if we are able to find two continuous functions f₁(x) and f₂(x) on [a, b] such that f₁(x) f(x) f₂(x) x [a, b], then

.

(iv) If m and M are respectively the global minimum and global maximum of f (x) in [a, b] then m (b - a) .

(v) , where f(x) and g(x) are two integrable functions.



Piecewise Continuous Functions

Consider a function f(x) defined on [a,b] which has discontinuities at finite number of points say x₁, x₂, ....,x_n. In such a case break the interval [a,b] into sub intervals such that f(x) is

continuous in each of the interval. Such a function is known as PIECEWISE continuous function. While calculating the definite integral of such a function we have to break the interval

[a,b] into sub intervals and calculate the integral separately in each of the sub intervals and add all of this to get the required answers .