## What is Application of Integration?

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

## Concept of Application of Integration

Basically, these calculations have been computed for ages. Both differentiation and integration have important roles in this matter. Application of integration is essentially useful for engineering, science, and mathematics. The major application of integration is done for the calculation of areas that cannot be solved by any other means.

Functions that impose problems such as finding out the areas under different types of curves use the application of integration. In a graphical representation, these areas are found out by taking the integral along the function. Used to find area that is formed between two curves, or an area that is formed due to the intersection of a curve and a line. In the discipline of mathematics, the application of integration plays a major role.

To find out the area under a curve, the total area is split into different columns of very small width. Then, the area of each of these columns is derived separately. Once the area of all the columns have been found out, then the sum of the areas is derived. The resultant sum is the area under the curve. This is nothing but integration, and this is the core concept of application of integration.

## What is Integral?

The application of integration is used to find out the integrals. The process of finding out the integrals can be termed as integration. When defining an integral, it also forms a function, deriving another given function.

The application integration is important, as it is used to find out the area of a region or volume of an object, whether it is a two-dimensional plane, or in a three-dimensional surface. While finding an integral where the function is integrated with respect to the variable ‘x’, then that means that the area between the curve and the x-axis. Another name of the integral is anti-derivative, as the process of finding out an integral is exactly the opposite of finding out a derivative.

## What are the Types of Integrals?

In application of integration, different types of integrals can be derived. The main two categories in which the integrals are divided are known as Definite integrals and Indefinite integrals.

### Definite Integrals

Definite Integrals is the process of integration that takes place between two fixed limits or intervals. This means that the addition takes places between two fixed points only. These two points are known as the upper limits and the lower limits respectively. Alternatively, definite integrals can also be termed as Riemann integral. It is generally denoted as-

$$\underset{a}{\overset{b}{\int}}f\left(x\right)dx$$Here, f(x) is the function that needs to be integrated, and ‘a’ and ‘b’ are the lower limit and the upper limit of the integral respectively.

### Indefinite Integrals

The term indefinite integral means finding out the integral of a function which does not have any lower or upper limits, or whose limits are not defined. Generally, represent an indefinite integral as

$$\int f\left(x\right)dx=F\left(x\right)+C$$Here, the term ‘C’ denotes a constant value that comes after the process of integration is complete. It is a legit addition of terms as if needed to derive the result, the constant term C will become zero on differentiation, making the function same as the one which was integrated.

## What is the Practical Application of Integrals?

With the application of integrals, there are various practical applications on which this process can be applied.

In Mathematics, the application of integration is used to find out the area that lies under any curve, the average value that is presented by a curve, an area that lies between any two curves, and to find out the centroid of an object that has curves on its sides. Different types of logarithmic functions and exponential functions cannot be estimated in terms of areas without the application of integration as well. The same goes for hyperbolic functions, which are solely based on integrable methods that can solve such related problems.

In Physics, the application of integration is useful for finding out the centre of mass, or the centre of gravity of an object. When an artificial satellite is placed on the orbit, then the satellite’s trajectory and velocity can be found out with the application of integration. The thrust of an object can be calculated with the application of integration too. Similarly, the momentum and mass of objects, whether moving or stationery, can be found out with the application of integration.

In Chemistry and Biology, used to study the rate of growth and decay in certain substances with compounds with the help of application integration. This has a special role in estimating the half-life of harmful radioactive substances, or finding out the duration of long-lost fossilized figures.

## Other Applications of Integration

Apart from these applications, application integration is necessary at the professional sector. Various companies need to integrate problems to get the proper solutions to their situations. As a result, companies are developing different software that will be able to integrate complex problems in a single platform. This way, the workers will be able to get the results quickly.

If an enterprise is running an integral problem that is based on their resource, then the accuracy and quickness of getting the results is prioritized. When software is created for running programs that can integrate complex functions, application integration is required for management.

## Formulas

- Common constant function is

The Integral is

ax + C

- Common variable function is

The Integral is

X^{2}/2 + C

- Common square function is

The Integral is

X^{3}/3 + C

- Common function for a reciprocal function is

The Integral is

In |X|+C

## Practice Problem

Solve the following integration

$$\underset{1}{\overset{2}{\int}}6\left({x}^{2}+x\right)dx$$solving the integration as follows

$$\begin{array}{l}{\displaystyle \underset{1}{\overset{2}{\int}}6\left({x}^{2}+x\right)dx}\\ =>6{\displaystyle \underset{1}{\overset{2}{\int}}\left({x}^{2}+x\right)dx}\\ =>6{\displaystyle \underset{1}{\overset{2}{\int}}{x}^{2}dx+}6{\displaystyle \underset{1}{\overset{2}{\int}}xdx}\\ =>6\frac{{\left[{x}^{3}\right]}_{1}^{2}}{3}+6\frac{{\left[{x}^{2}\right]}_{1}^{2}}{2}\\ =>6\frac{\left({2}^{3}-{1}^{3}\right)}{3}+6\frac{\left({2}^{2}-{1}^{2}\right)}{2}\\ =>2(8-1)+3(4-1)\\ =>2\ast 7+3\ast 3\\ =>14+9\\ =>23\end{array}$$This is the required answer.

## Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

- B.Sc. in Mathematics
- B.Sc. in Physics
- M.Sc. in Mathematics

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