Page 296 - DMTH401_REAL ANALYSIS

Real Analysis

Notes According to this theorem, differentiation and integration are inverse operations.
We now discuss a theorem which establishes the required relationship between differentiation
and integration. This is called the Fundamental Theorem of Calculus.

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.

Theorem 3: Fundamental Theorem of Calculus
tb
ò
-
If f is integrable on [a,b] and F is a primitive of f on [a,b], then f(x) dx = F(b) F(a).
a
b
Proof: Since f Î R [a,b], therefore limS(P,f) = ò f(x) dx
P 0
-
a
where P = {x , x , x ,...., x ] is a partition of [a,b]. The Riemann sum S(P,f) is given by
0 1 2 n
n n
D
S(P,f) = å f(t ) x = å f(t )(x - x i 1 ); x - £ t £ t .
1
-
i
i
i
i
i
i
i
=
=
i 1 i 1
Since F is the primitive of f on [a, b], therefore F' (x) £ f(x), x Î [a, b].
n
Hence S(P,f) = å F'(t )(x - x i 1 ). We choose the points t, as follows:
i
-
i
i 1
=
By Lagrange's Mean Value theorem of Differentiability, there is a point t, in ]x , x [ such that
i-1 i
F(x ) – F(x ) = F' (t ) (x, – x )
i i-1 i i-1
n
=
-
Therefore, S(P,f) = å [F(x ) F(x i 1 )] F(x ) F(x ) F(b) F(a).
-
=
-
i
0
n
-
i 1
=
b
Take the limit as P  0. Then f(x) dx = F(b) F(a). This completes the proof.
-
ò
a
Alternatively, the Fundamental Theorem of Calculus is also interpreted by stating that the
derivative of the integral of a continuous function is the function itself.
b
If the derivative f of a function f is integrable on [a, b], then f'(x) dx = f(b) f(a).
ò
-
a
Applying this theorem, we can find the integral of various functions very easily.
Consider the following example:
t
ò
Example: Show that sinx dx = 1 cost.
-
0
Solution: Since g(x) = – cos x is the primitive of f(x) = sin x in the interval [0, t], therefore
t
ò Sin x dx = g(t) g(o) = 1 cost.
-
-
0
b
We have, thus, reduced the problem of evaluating f(x) dx to that of finding primitive of f on
ò
a
b
[a, b]. Once the primitive is known, the value of f(x) dx is easily given by the Fundamental
ò
a
Theorem of Calculus.
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