Applications of Integrals in Mathematics

applications of integrals

Applications of integrals can be tricky- it can be applied to math, science, and engineering, economics etc. If you don’t have the right teacher or textbook survival guide, getting lost before exam can be easy.

In this guide you’re going to learn how integration is applied in various scenarios, the different types of integrals, and some examples. 

Applications of Integrals: What does it mean?

Integrals can be applied in many different areas like math, science, and engineering, statistics,and economics.

 In math, we use it to find:

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  • The center of mass (Centroid) of an area with curved sides
  • The area between two curves
  • The area under a curve
  • The average value of a curve
  • Integral transforms like Laplace and Fourier transforms. 

When speaking about integrals, there are two different types: definite integrals (also called Riemann Integral) and indefinite integrals.

In simple terms, a definite integral has definite limits ( defined upper and lower limits/bounds) which can be represented by:

definite integral

In contrast to definite integrals, an indefinite integral is an integral which has no limits. This can be represented by:

indefinite integral

                                               (C is the constant of integration)

Solved: Example Problem of Applications of Integrals

applications of integrals problem

The given function is a type of parabola, the graph is as shown below:

applications of integrals graph

First, take a look at our notes below:

definite integrals formula
applications of integrals problem
applications of integrals problem

We will integrate the given function to get . We know that using which we will compute the value of the constant of integration. Once we know the value of constant of integration, the next step is to find f(4).

applications of integrals problem

1) Knowing all that, let’s integrate .To do that, we need to apply the power rule of integration which would be:

power rule of integration

2)  Using the power rule, 

applications of integrals problem
applications of integrals problem

                                                  ……………(1)

Thus we got the result of integration.

3) Now we will find the value of the constant of integration, 

applications of integrals problem

We know that , hence applying this value to equation(1) we get

applications of integrals problem

                                                  

applications of integrals problem

                                                  

applications of integrals problem

                                                  

applications of integrals problem

                                                  

applications of integrals problem

                                                       

applications of integrals problem

Therefore we found the value of the constant of integration which is

constant of integration

(4) Now we will apply this value of integration constant to

applications of integrals problem
applications of integrals problem

Conclusion

This is just one example of many different scenarios. We have to remember in math alone there are numerous applications and in physics even more! Applications of integrals is a topic that takes practice with the right survival guides and experts to help you with solutions– you’ll be on the right track.

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