simple explanation on fundamental theorem of definite antiderivative | significance of definite integral | Knowledge of Physics

 It has been discussed HERE that if F(x) is a continuous function of x on interval [a,b] then f(x) is antiderivative of F(x) if

`\frac{d}{dx} F(x) = f(x)` ----(1) 

The definite integral of f(x) on the interval [a,b] is defined as

`\int_a^b f(x) dx = [F(x)]_a^b = F(b) - F(a)` -----(2)

Here, a = lower limit of x

         b = upper limit of x

Two points to be remembered about definite integral,

1. limits of the variable should be defined.
2. definite integral has definite value

Fundamental Theorem of Integral Calculus:

The theorem which is applicable to determine the definite integral value of given function on given interval is called fundamental theorem of integral calculus.

This theorem establishes the relation between two basic concepts of calculus: the derivative and the definite integral.

Statement: If F(x) is continuous on [a,b] and f(x) is an antiderivative of F(x), then

`\int_a^b f(x) dx = F(b) - F(a)` 

Proof:

 

Let, `F(x) = \int_a^b f(x) dx` ----(1)

Suppose, `F(x)` and `\Phi`(x)  are antiderivatives of same function `f(x)`. Then these antiderivatives differ by a constant. That is

`F(x) - \Phi(x) = constant`

or, `F(x) - \Phi(x) = c` 

`\therefore` `F(x) = \Phi(x) + c`

For `F(a) = 0`,

`0 = \Phi(a) +c`

`\Phi(a) = -c` ------(2)

Also,

`F(x) = \Phi(x) + c` 

For x = b

`F(b) = \Phi(b) + c`

or, `\F(b) = \Phi(b) - \Phi(a)` `\therefore` by using (2)

`\therefore` `F(b) = \Phi(b) - \Phi(a)` -----(3) 

From equation (1) and (3),

`\int_a^b f(x) dx = \Phi(b) - \Phi(a)`

Hence, `\int_a^b f(x) dx = \Phi(b) - \Phi(a)`

In general,

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

This proves the theorem.

Significance of Antiderivative:

Suppose, F(x) is a function continuous on interval `x \epsilon  [a,b]`, then F(x) is antiderivative f(x) if the derivative of F(x) is f(x). That is,

`\frac{d}{dx} F(x) = f(x)`

Fig.1 Significance of definite integral of function on given interval

The integral value of f(x) at x = a, x = b gives the area of the surface enclosed by the curve represented by f(x), x = a, x = b and x-axis, as shown in fig.1. This is the major significance of the integral value of a function. That is, the area of the surface enclosed within ABCD in above figure is given as

Area of ABCD = `\int_a^b f(x) dx = F(b) - F(a)` -----(4)

Similarly,

Definite integral is also used for finding volume enclosed by the given curves and limiting values. Thus definite integral has a great importance in our day-to-day life. This is mostly useful in the fields of physics, statistics, engineering etc.

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