Definite Integral





Definite Integral:

Definite integral is a form of Integral or Anti derivative in which we don’t get a range of answer or indefinite answer , Instead we get a fixed or definite answer.

Or, A definite integral is the integral of a function in a closed interval and it is denoted by:

\int^b_a f(x) . d(x)

Which means , the Integral or Anti derivative of the function “f(x)” in an interval from “a” to “b”.

Definite Integral Formula:

We can calculate or evaluate a definite integral using the definite integral formula which states that:

If “f” is continuous on [a,b] and \phi is any antiderivative of “f” then ,

\int^b_a f(x) . dx = \phi (b) - \phi (a)

Now let us prove the formula:

We can prove the formula using the fundamental theorem of calculus which states that:

If “f” is continuous function and F(x) = \int^x_a f(t) .dt  , then ,

\dfrac{d}{x} F(x) = f(x)

Now let us prove the definite integral formula with the help of fundamental theorem of calculus,

Proof:

Let,  F(x) = \int^x_a f(t) .dt

As “a” is a constant obviously we get “F(a) = 0″.

And as, “F” and \phi are antiderivatives of same function “f” , they differ by a constant “c”  as stated in antiderivatives .

So,

F(x) = \phi (x) + c For some constant “c”.

Thus , F(a) = \phi (a) + c

or, 0 = \phi (x) + c

or , \phi (x) = - c

So ,

F(b) = \phi (b) - \phi (a)

And we also have,

F(b) = \int^b_a f(t) .dt

Thus , \int^b_a f(t) . dt = \phi (b) - \phi (a)

Example of Definite Integrals:

We can evaluate definite integrals using the same techniques of integration we used while evaluating indefinite integrals.

One example of definite integral is:

\int^3_0 x^5 .dx

We can easily evaluate this integral using Integration Formulas .

So we get:

\int^3_0 x^5 .dx = \left( \dfrac{1}{6} x^6 \right)^3_0

= \frac{1}{6} ( 3^6 - 0^6)

= \frac{243}{2}



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