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Applications of Integrals

There is a number of methods of calculations among which are functions, differentiation, and integration. Applications of Integrals are applied in various fields like Mathematics, Science, Engineering. Further, for the calculation of areas or irregular shapes in a two-dimensional space, we use majorly integrals formulas.

Here a brief introduction on integrals is given, with applications of integrals to find areas under simple curves, areas bounded by a curve and a line and area between two curves, and also the application of integrals in other mathematical disciplines along with the solved examples.

1.	Definition of Integral
2.	Types of Integrals
3.	Applications of Integrals
5.	Solved Examples on Applications of Integrals
6.	Practice Questions on Applications of Integrals
7.	FAQs on Applications of Integrals

Definition of Integral

Given the derivative f’ of the function f, a question that arises is, "Can we determine the function f?" Here, the function f is called antiderivative or integral of f’. The process of finding the antiderivative is called integration. On the other hand, the value of the function found by the process of integration is called an Integral.

For example, the derivative of f(x) = x³ is f’(x) = 3x²; and the antiderivative of g(x) = 3x² is f(x) = x^3. Here, the integral of g(x) = 3x² is f(x)=x³

Definition of integral: An integral is a function, of which a given function is the derivative. Integration is basically used to find the areas of the two-dimensional region and for computing volumes of three-dimensional objects. Therefore, finding the integral of a function with respect to the x-axis refers to finding the area of the curve with respect to the x-axis. The integral is also called as anti-derivative as it is the reverse process of differentiation.

In general, there are two types of integrals. Definite integrals are defined for integrals with limits and indefinite integrals do not include any limits. Here, let us explore more about definite, and indefinite integrals.

Types of Integrals

Definite Integrals

These are the integrals that have a pre-existing value of limits; thus making the final value of integral definite. The definite integrals are used to find the area under the curve with respect to one of the coordinate axes, and with the defined limits. Here we aim at finding the area under the curve g(x) with respect to the x-axis and having the limits from b to a.

Indefinite Integrals

These are the integrals that do not have a pre-existing value of limits; thus making the final value of integral indefinite. The indefinite integrals are used to integrate the algebraic expressions, trigonometric functions, logarithmic, and exponential functions. Here g'(x) is the derivative answer, which on integration results in the original function of g(x). The integration does not give back the constant value of the original expression, and hence a constant 'c' is added to the answer of the integral.

indefinite integrals

Application of Integrals

From the many applications of integrals, some are listed below:

In Mathematics integrals are used to find:

Center of mass(Centroid) of an area having curved sides
The average value of a curve
The area between two curves
The area under a curve

In Physics integrals are used to find:

Centre of gravity
Center of mass
Mass and moment of inertia of vehicles
Mass and momentum of satellites
The velocity and trajectory of a satellite
Thrust

There are two forms of integrals. Indefinite Integrals: It is an integral of a function when there is no limit for integration. It contains an arbitrary constant. Definite Integrals: An integral of a function with limits of integration.

Application of integrals also includes finding the area enclosed in the eclipse, the area of the region bounded by the curve, or any enclosed area bounded in the x-axis and y-axis. The application of integrations varies depending upon the fields. Graphic designers use it for the creation of three-dimensional models. Physicists use it to determine the center of gravity, etc.

Let us have a look at one of the common applications of integrals i.e., how to find area under the curve.

How to Find Area Under The Curve?

The area under the curve can be calculated through three simple steps. First, we need to know the equation of the curve(y = f(x)), the limits across which the area is to be calculated, and the axis enclosing the area. Secondly, we have to find the integration (antiderivative) of the curve. Finally, we need to apply the upper limit and lower limit to the integral answer and take the difference to obtain the area under the curve.

Area = \(_a\int^b y.dx \)

= \(_a\int^b f(x).dx\)

=\( [g(x)]^b_a\)

=\( g(b) - g(a)\)

Area Under The Curve

Related Topics:

Important Notes on Applications of Integrals:

The value of the function found by the process of integration is called an integral.
In general, there are two types of integrals:
Definite Integrals (the value of the integral is definite)
Indefinite Integrals (the value of the integral is indefinite)

Solved Examples on Applications of Integrals

Example 1: Can you find the area under the curve using the application of integrals, for the region bounded by the circle x² + y² = 16 in the first quadrant?

Solution:

The given equation of the circle is x² + y² = 16
Simplifying this equation we have \(y=\sqrt {4^2-x^2}\)

Here we find the area of the quadrant of the circle across the limits [0, 4] and then multiply it by 4 to obtain the area of the circle.
\(\mathrm{A}=4\int_{0}^{4} y \cdot d x\)
\(=4\int_{0}^{4} \sqrt{4^{2}-x^{2}} \cdot d x\)
\(=4\left[\frac{x}{2} \sqrt{4^{2}-x^{2}}+\frac{4^{2}}{2} \operatorname{Sin}^{-1} \frac{x}{4}\right]_{0}^{4}\)
\(=4[((4 / 2) \times 0+(16 / 2) \operatorname{Sin}-11)-0]\)
\(=4(16 / 2)(\pi / 2)\)
\(=16 \pi\)

Answer: Therefore the area of the region bounded by the circle in the first quadrant is 16π sq units.
Example 2: Find the area under the curve using the application of integrals, for the region enclosed by the ellipse x²/36 + y²/25 = 1.

Solution: The given equation of the ellipse is x²/36 + y²/25 = 1
This can be transformed to obtain \(y=\frac{5}{6} \sqrt{6^{2}-x^{2}}\)
\(\begin{aligned} A &=4 \int_{0}^{6} y \cdot d x \\ &=4 \int_{0}^{6} \frac{5}{6} \cdot \sqrt{6^{2}-x^{2}} \cdot d x \\ &=\frac{20}{6}\left[\frac{x}{2} \cdot \sqrt{6^{2}-x^{2}}+\frac{6^{2}}{2} \sin ^{-1} \frac{x}{6}\right]_{0}^{6} \\ & \left.=\frac{20}{6}\left[\left(\frac{6}{2} \times 0\right)+\frac{6^{2}}{2} \cdot \operatorname{Sin}^{-1} 1\right)-0\right] \\ &=\frac{20}{6} \cdot \frac{36}{2} \cdot \frac{\pi}{2} \\ &=30 \pi \end{aligned}\)

Answer: Therefore the area of the ellipse is 30π sq units.

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Practice Questions on Applications of Integrals

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FAQs on Applications of Integrals

What Are the Applications of Integrals in Real Life?

Integration finds many uses in the fields of Engineering, Physics, Maths, etc. For example in Physics, Integration is very much needed. For example, to calculate the Centre of Mass, Centre of Gravity, and Mass Moment of Inertia of a sports utility vehicle. To calculate the velocity and trajectory of an object, predict the position of planets, and understand electromagnetism.

What Are the Real-Life Applications of Integrals and Differentiation?

Differentiation and integration can help us solve many types of real-world problems. We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.). And the applications of integrals are useful to find the areas of irregular shapes.

What Is the Application of Integrals in Maths?

Integrals are used to evaluate such quantities as area, volume, work, and, in general, any quantity that can be interpreted as the area under a curve.

What Is Integration in Simple Words?

In Maths, integration is a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation, where we reduce the functions into parts. This method is used to find the summation under a vast scale.

How Do You Define Integration?

An integral is a function, of which a given function is the derivative. Integration is basically used to find the areas of the two-dimensional region and computing volumes of three-dimensional objects. Therefore, finding the integral of a function with respect to x means finding the area with respect to the X-axis and the curve. The integral is also called as anti-derivative as it is the reverse process of differentiation.

What Are the Two Types of Integrals?

There are two forms of integrals.

Indefinite Integrals: It is an integral of a function when there is no limit for integration. It contains an arbitrary constant.
Definite Integrals: An integral of a function with limits of integration.

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