Definite Integral

posted on: 19 Jan, 2012 | updated on: 25 Aug, 2012

An integral with upper and lower limits is known as definite integral. We can represent a Definite Integral as shown below-

∫_p^q f ( x ) dx.

Here p, q, and x are complex Numbers.

We calculate the definite integrals in terms of indefinite integrals, let F be the indefinite integral for a continuous function f ( x ). Therefore it is calculated as-

∫_p^qf ( x ) . dx = F ( q ) - F ( p )

So when we evaluate the definite integral then first thing we do is calculate the indefinite integral and then apply limits. Our function should be continuous in the interval of Integration; this is most important property of a definite integral. We need to keep above things in mind to evaluate the definite integral-

Let us take an example to understand the concept of definite integrals. The example is shown below-

Example 1) We have a integral ∫₀² ( x² + 1 ) dx

Solution) So after integrating, we get-

F ( x ) = [ x³/ 3 + x ] ₀²

= [ ( 2 ) ³ / 3 + 2 ] ^{- 0}

= 8 / 3 + 2,

= 14 / 3,

Example 2) Find the definite integral of ∫₂³ ( x³ + x² ) dx.

Solution) After integrating the above function we get-

=> F ( x ) = [ x4 + x3 ] ₂ ³

Now applying limits, we get-

=> F ( 3 ) - F ( 2 ),

=> F ( 3 ) - F ( 2 ) = [ ( 3 ) ⁴ + ( 3 ) ³ ] - [ ( 2 ) ⁴ + ( 2 )³ ],

=> F ( 3 ) - F ( 2 ) = [ 108 – 27 ],

= 84.

So this is how we evaluate definite integrals.

Topics Covered in Definite Integral

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posted on: 19 Jan, 2012 | updated on: 25 Aug, 2012

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