Class 12 Maths Integrals Fundamental Theorem of Calculus

Fundamental Theorem of Calculus

There are some fundamental theorems which are going to discuss here one by one.

Area function

Let x be the given point in [a, b] as shown in the figure.

Then aʃb f(x) dx represents the area of the shaded region. The function A(x) is represented as Area function and is calculated as

A(x) = aʃb f(x) dx

According to this definition, two fundamental theorems are stated as:

First fundamental theorem of integral calculus

Theorem 1: Let f be a continuous function on the closed interval [a, b] and let A (x) be the area function.

Then A′(x) = f (x), for all x ∈ [a, b].

Second fundamental theorem of integral calculus

Theorem 2: Let f be continuous function defined on the closed interval [a, b] and F be an anti derivative of f. Then aʃb f(x) dx = [F(x) a]b = F(b) – F(a), where F(x) is the indefinite integral of f(x).

Problem: Evaluate the following definite integrals

(a) ʃ-11 (x + 1) dx                                                         (b) ʃ21 dx/x

Solution:

(a) Let I = ʃ-11 (x + 1) dx

Now, ʃ (x + 1) dx = x2/2 + x = F(x)

By second fundamental theorem of calculus, we get

I = F(1) - F(-1)

= (1/2 + 1) – (1/2 - 1)

= 1/2 + 1 – 1/2 + 1

= 2

So, ʃ-11 (x + 1) dx = 2

(b) Let I = ʃ21 dx/x

Now, ʃ dx/x = log|x| = F(x)

By second fundamental theorem of calculus, we get

I = F(3) - F(2)

= log|3| - log|2|

= log 3 – log 2

= log(3/2)

So, ʃ21 dx/x = log(3/2)

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