Definite Integral | Mathematics

Definite integrals are the extension after indefinite integrals, definite integrals have limits [a, b]. It gives the area of a curve bounded between given limits.

\int_{a}^{b}F(x)dx, It denotes the area of curve F(x) bounded between a and b, where a is the lower limit and b is the upper limit.

Note: If f is a continuous function defined on the closed interval [a, b] and F be an anti derivative of f. Then \int_{a}^{b}f(x)dx= \left [ F(x) \right ]_{a}^{b}\right = F(b)-F(a)
Here, the function f needs to be well defined and continuous in [a, b].



Example: Find, \int_{1}^{4}x^{2}dx ?

Solution:

Since, \int x^{2}=\frac{x^{3}}{3}  \newline \newline \textup{Then F(x)} =\frac{x^{3}}{3} \newline \newline [F(x)]_{1}^{4}= F(4)-F(1) \newline \newline =[\frac{4^{3}}{3} - \frac{1^{3}}{3}]=\frac{65}{3}
 

    Properties of definite integrals –

  1.  \int_{a}^{b}f(x)dx=\int_{a}^{b}f(t)dt

  2. \int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx

  3. \int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx

  4. \int_{a}^{b}f(x)=\int_{a}^{b}f(a+b-x)dx

  5. \int_{0}^{b}f(x)=\int_{0}^{b}f(b-x)dx

  6. \int_{0}^{2a}f(x)dx=\int_{0}^{a}f(x)dx+\int_{0}^{a}f(2a-x)dx

  7. \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx, \textup{if f(x) is even function i.e f(x)=f(-x)}

  8. \int_{-a}^{a}f(x)dx=0, \textup{if f(x) is odd function}

These properties can be used directly to find the value of particular definite integral and also interchange to other forms if required.



My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.




Article Tags :

Be the First to upvote.


Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.