Thursday, December 13, 2012

Fundamental Theorem of Integral Calculus

In this section we are going to discuss about the fundamental theorem of integral calculus concept. The development of short form the fundamental theorem of integral calculus corresponding to over screening the significant concept and problem in using calculus theorems are referred as review calculus. This article helps to improve the knowledge for using fundamental theorem of integral calculus problem and below the problems are helping toll for the exam. Fundamental theorem of integral calculus problem solutions also shows below. The fundamental theorem of integral calculus handled the differentiation, integration and inverse operations are process here now.

Important of Fundamental Theorem of Integral Calculus:-

Let` f(x)` is a continuous function on the closed interval `[a, b]` .

Let the area function `A(x)` be defined by `A(x) = int_a^xf(x)dx ` for` xgt=a`

Then `A'(x) = f(x)` for all `X in [a,b]`

Let` f(x)` be a continuous function defined on an interval `[a,b]` .

`If intf(x)dx = F(x) then int_a^a f(x)dx = [F(x)]^b_a`

`=F(b) - F (a)` is called the definite integral or `f (x) ` among the limits` a` and `b` .

This declaration is also known as fundamental theorem of calculus.

We identify `b` the upper limit of `x` and a the lower limit.

If in place of `F(x)` we take` F(x) +c` as the value of the integral, we have

`int_a^b f(x)dx = [F(x) + C ]^b_a`

`= [F (b) + c] - [F (a) + c]`

`= F (b) + c - F (a) - c`

`= F (b) - F (a)`

Therefore, the value of a definite integral is unique. It does not depend on the constant c and hence in the evaluation of a definite integral the constant of integration does not play any role.

Let `int f(x) dx = F(x) +C`

Then `int_a^b f(x)dx = F(b)-F(a)`

Note down:-

From the above two theorem, we infer the following

`intf(x)dx` =(Anti derivative of the function `f(x) ` at `b` ) - (Anti derivative of the function `f(x) ` at `a` )

The fundamental theorem of integral calculus gives you an idea about a close relationship between differentiation and integration. These theorems give an exchange method evaluating definite integral, without calculating the limit of a sum.


Example on Fundamental Theorem of Integral Calculus:-


Evaluate the definite integral of the following

`int^(pi/4)_0(3sec^2 x + x^2 + 3)`

Solution:

`int_0^(pi/4) (3sec^2 x + x^2 + 3)`

`= 3 int_0^(pi/4)sec^2 xdx + int_0^(pi/4)x^2dx + int_0^(pi/4)3dx`

`= 3[tanx] ^ (pi/4) _0 + [(x^3)/ (3)] ^ (pi/4) _0 + [3x] ^ (pi/4) _0`

`= 3 (tan (pi/4) - tan0) + 1/3 (pi/4) ^3 - 0+ [(pi/2) - 0]`

`= 3 + (pi^3)/192+ (pi)/ (2).`

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