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 q f ( 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 ],
So this is how we evaluate definite integrals.
The Fundamental Theorem of Integral Calculus is one of the most interesting thing that one after having done with lot of differentiation and Integration concepts would love to read. This Theorem relates the concepts of differentiation and integration too closely, which helps the users to understand its concepts more predominantly.
If f is ...Read More
The Riemann integral was developed by Bernhard Riemann. So the name of formula is Riemann integral on the bases of developer name. Bernhard Riemann gives the definition of the integral in Calculus where the function has interval. This formula is mostly applicable in the practical purpose and it is a rare case that we can apply i...Read More
Find the area of the region bounded by interval in Integration is one of the important topics of integration Calculus. In calculus when two Functions are intersect each other and the area lie between the intervals then it means the area is bounded by interval. Suppose we have two function if we take the interval between to two function is [a, b] then formula o...Read More
To understand definite Integration in the Calculus, we have to first go through the definition of the integration, it is the opposite process of the differentiation and always used to find the sum of areas of small parts of any irregular and big body for example an unknown field. We have generally two types of the integrals in calculus. First type of i...Read More
A Definite Integral is an integral with limits. The definite integral is of the form ∫ ab f( x ) dx where a, b, and x can be complex Numbers. The definite integral can be also be defined as- Let f( x ) be...Read More