Definite Integrals





This page definite integrals we are going to see the definition of definite- integral and also example problems using limit.

Definition:

A basic concept of integral calculus is limit. Generally the concept integration is used to find area between curves within certain limit.

Example 1

Evaluate the following 

Solution:

To solve this problem we have to use substitution method. That is we are going to change the given function from one variable to another variable. So let us consider

t = cos x

differentiating both side with respect to x

          dt = - sin x dx 

- sin x dx = dt

 sin x dx = - dt

We have changed the given function in terms of "t" from the variable x .So we need to change the limits also.

 When x = 0                  when x = Π/2

         t = cos 0                     t = cos Π/2

         t = 1                           t = 0 



Example 2

Evaluate the following        

Solution:

To solve this problem we have to use the trigonometric formula for sin² x.

The formula for sin² x is (1 - cos 2x)/2

In the first step we have applied the trigonometric formula for sin² x. In the second step we have taken 1/2. Now we got integral (1-cos 2x) .If we integrate 1 we will get x and if we integrate cos 2x we will get sin 2x/2. Then we have applied the upper limit first and then lower limit.


Example 3

Evaluate the following 


Solution:

To solve this problem we have to use substitution method. That is we are going to change the given function from one variable to another variable. So let us consider

 t = Sin⁻¹ x

 differentiating with respect to x on both sides

 dt = 1/√(1-x²) dx

 We have changed the given function in terms of "t" from the variable x .So we need to change the limits also.

 When x = 0                  when x = 1

        t = Sin⁻¹ 0                     t = Sin⁻¹(1)

         t = 0                           t = Π/2  



Related pages

Quote on Mathematics definite integrals  definite integrals

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are: 

It subtracts sadness and adds happiness in our life.    

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”

HTML Comment Box is loading comments...