Page 298 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 24.3 Summary

 The main thrust of this unit has been to establish the relationship between differentiation
and integration with the help of the Fundamental Theorem of Calculus.

 We have discussed some important properties of the Riemann Integral. We have shown
that the inequality between any two functions is preserved by their corresponding Riemann
integrals; the modulus of the limit of a sum never exceeds the limit of the sum of their
module and if we split the interval over which we are integrating a function into two
parts, then the value of the integral over the whole will be the sum of the two integrals
over the subintervals.
 Let f: [a, b]  R be a continuous function. Let F : [a, b]  R be a function defined by
x
F(x) = ò f(t) dt, x E[a,b].
a
Then F'(x) = f(x), a £ x £ b.

This is the first result which links the concepts of integral and derivative. It says that, if f is
continuous on [a, b] then there is a function F on [a, b] such that F'(x) = f(x)," x [a,b].
Î
You have seen that if f: [a, b]  R is continuous, then there is a function F: [a, b]  R such
that F' (x) = f(x) on [a, b]. Is such a function F unique? Clearly the answer is 'no'. For, if you
add a constant to the function F, the derivative is not altered. In other words, if
x
c
G(x) = + ò f(t) dt for a £ x £ b then also G' (x) = f(x) on [a, b].
1
 It states that the integral of the derivative of a function is given by the function itself.
 The Fundamental Theorem of Calculus was given by an English mathematician Isaac
Barrow [1630-1677], the teacher of great Isaac Newton.
 The following observations are obvious from the theorems 1 and 2.
(i) If f is integrable on [a, b], then there is a function F which is associated with f through
the process of integration and the domain of F is the same as the interval [a, b] over
which f is integrated.
(ii) F is continuous. In other words, the process of integration generates continuous
function.
(iii) If the function f is continuous on [a, b], then F is differentiable on [a, b]. Thus, the
process of integration generates differentiable functions.
(iv) At any point of continuity of f, we will have f(c) = f(c) for c e [a, b]. This means that
if f is continuous on the whole of [a, b], then F will be a member of the family of
primitives of f on [a, b].

24.4 Keywords

Primitive of a Function: If f and F are functions defined on [a, b] such that F’(x) = f(x) for x Î [a, b]
then F is called a 'primitive' or an 'antiderivative' off on [a, b].
Fundamental Theorem of Calculus: If f is integrable on [a, b] and F is a primitive of f on [a, b],
tb
ò
then f(x) dx = F(b) F(a).
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a

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