Application of Integrals

Application of Integrals are used to find the area and volume of various 2-D and 3-D curves and they have vast applications in the fields of mathematics and physics. They generally help us to calculate the area of the curve, irregular contour, the volume of various curves, and others.

In this article, we will discuss Applications of Integral and its meaning. We will also understand its application and formulae to calculate the Integral value in mathematics. We will also solve various examples and provide practice questions for a better understanding of the concept of this article. We have to study the Application of Integrals in Class 12.

Application of Integral in Mathematics

Integral involves the summation of discrete data and its applications cover the basic principles of integrals, including the fundamental theorem of calculus. In mathematics, the Application of integral is used to determine the area under a curve and the area between two curves.

The diverse Applications of Integrals extend into mathematics, physics, and real-world situations, where they play an important role in calculating areas, volumes, and displacements resulting from the collection of non-measurable data points.

For the study of calculus, the concept of limit is required. While simple geometry equations are used to calculate the areas of straight and simple shapes, the concept of integral calculus becomes essential when dealing with areas enclosed by curves.

Definition of Integrals

An Integral is defined as a function that corresponds to the antiderivative of a given function. Its primary aim is to determine the areas of two-dimensional space and calculate volumes for three-dimensional objects.

Integral is the representation of calculating the area under a curve with respect to the x-axis which is equivalent to the integral of a function with respect to the x-axis. The process of finding out the anti-derivative is known as integration, and the numerical value derived from this process is termed an Integral.

Types of Integrals

There are two broad types of Integrals: Definite Integrals and Indefinite Integrals. Definite Integrals are defined for integrals with limits and Indefinite Integrals do not include any limits.

Definite Integrals

Definite Integrals are defined as the calculation of antiderivatives over specific intervals or fixed boundaries. It is also known as Riemann Integral. It represents the area bounded by a curve with lower limit ‘a’ and upper limit ‘b’ for a given function f(x) with respect to the x-axis. The representation is given by

∫_a^b f(x).dx = F(b) -F(a), where a ≤ x ≤ b

Indefinite Integrals

Indefinite Integrals do not have any predefined limits of integration i.e. its upper and lower limits are not defined. It integrates a variety of functions, including algebraic expressions, trigonometric functions, logarithmic, and exponential functions.

In the case of indefinite integrals, z′(x) represents the derivative solution of any given function z(x), the integration of z′(x) gives back the original function z(x). This integration alone does not provide the original expression, a constant ‘C’ is added to the integral result. The Indefinite Integral is defined as

∫z(x).dx = Z(x) + C

Where,

• Derivative of Z(x) is the Original Function z(x)
• C is the Constant of Integration
• Z(x) is the Value of Indefinite Integral

Application of Integrals

There are various Applications of Integrals in mathematics. Some are mentioned below:

• It is used to calculate the area under simple curves.
• Find areas enclosed by lines, arcs of circles, parabolas, ellipses.
• Find out the centroids of areas of Triangle with curved boundaries.
• Find out the area between two curves and more.

How to Find Area Between Curve and Axis?

We can find the Area between a curve y = f(x), the x-axis, and specific intervals that is the lines x = a and x = b by using integration:

∫_a^b y.dx = ∫_a^b f(x).dx = F(b) -F(a)

Similarly, when dealing with the region enclosed by the curve x= g(y), the y-axis, and the lines y = a and y = b, the Integral expression is:

∫_a^b x.dy = ∫_a^b g(y).dy = G(b) -G(a)

How to Find Area Between Two Curve?

For areas between two curves y = g(x) and y = f(x), where f(x) ≥ g(x) in the interval [a, b], the area between x = a and x = b is:

∫_a^b f(x).dx – ∫_a^b g(x).dx = ∫_a^b {f(x) – g(x)}.dx

Similarly, for regions between two curves x = g(y) and x = f(y), where f(y) ≥ g(y) in the interval [c, d], the Integral expression becomes:

∫_a^b f(y).dy – ∫_a^b g(y).dy = ∫_a^b {f(y) – g(y)}.dx

Learn more about, Area Between Two Curves.

How to Find Area Under Curve?

To calculate the Area under a Curve, follow the steps added below

• Step1: Firstly identify the equation of the curve y=f(x), the limits and the axis for area calculation.
• Step2: The integration (antiderivative) of the curve is found.
• Step3: The upper and lower limits are applied to the integral result, and the difference gives the area under the curve:

Area = ∫_a^b y.dx

⇒ Area = ∫_a^b g(x).dx

⇒ Area = [g(x)]^b_a

Area = g(b) − g(a)

Application of Integrals in Physics

Various applications of Integral in Physcis are,

• Integrals are used to calculate the center of mass for objects like cars, buildings etc to acquire stability.
• It is used to calculate the Center of Gravity and moment of inertia for vehicles and satellites. It is used to understand the stability and rotational characteristics of these objects.
• Integrals are used to calculate work done, where the force is a function of displacement.
• It is used in fluid mechanics for the calculation of flow rate, volume, and pressure distribution.
• Integrals are used to predict the trajectory of a satellite during its orbital placement to ensure it follows the desired path or not.
• Integrals are applied to calculate thrust.

Application of Integral in Engineering and Mathematics

Various applications of Integral in Engineering and Mathematics are,

• Integration is used to estimate the area under a curve in a given interval.
• Integration can be used for multiple Integrals not only for single-variable functions but also for double and triple integrals.
• Integral calculus can be used in curve tracing for mapping and analyzing the paths of functions in both Cartesian and polar coordinates.
• Integrals can be used in length estimation of a curve and calculate the volume of a solid formed by revolving a curve around an axis.
• Integral calculus is used to estimate the amount of material needed for curved surfaces in any construction works.
• Integral is used in calculating the length of power cables required for proper transmission between two power stations.
• Integrals can be used in computer science in image processing for edge detection and feature extraction.

Application of Integrals in Different Fields

Various applications of Integral in Different Fields are,

• Integrals are used in medical science to understand bacterial growth dynamics and food conditions.
• In statistics, Integrals play a role in evaluating survey data.
• Integrals are used to estimate the population sizes over time and predict the future trends of population.

Related Resources

• Integration in Maths
• Integration Formulas
• Antiderivatives

Examples on Applications of Integrals

Example1: Determine the area enclosed between the curve represented by y² =4x and the linear function y=x.

Solution:

Given,

• x² = 4x

Rearrange it to obtain a quadratic equation:

x² − 4x = 0

x(x−4) = 0

x = 0 and x = 4

Now, use the definite integral for the calculation of area between the curve and the line:

Area = ∫₀⁴ (y – y²/2).dx

Here, y = x is the upper curve, and y² = 4x is the lower curve.

Area = ∫₀⁴ (x – x²/4).dx

Now, integrate with respect to x,

Area = [x²/2 − x³/12]₀⁴

Area = (4²/2−4³/12) − (0²/2−0³/12)

Area = (8−64/12) − (0−0)

Area = (8−16/3)

Area = 24/3 − 16/3 = 8/3

Therefore, the Area of the region enclosed between the curve y² = 4x and the straight line y = x is 8/3 square units.

Example 2: Determine the area enclosed between the curve represented by y² =4y-x and y axis.

Solution:

Write the curve equation in terms of y: y² = 4y−x

y² − 4y + x = 0

Put x in terms of y: x = y² − 4y

Put x=0 and solve for y: 0 = y² − 4y

y(y−4) = 0

So, y = 0 and y = 4 are the points where the curve intersects the y-axis.

Area between the curve and the y-axis is given by the integral,

A = ∫_y1^y2 x.dy

In this case, y₁ = 0 and y₂ = 4

A = ∫₀⁴ (y² − 4y).dy

A = [y³/3−2y²]₀⁴

A = (4³/3−2(4²)) − (0³/3−2(0²))

A = -32/3

Example 3: The area bounded by the curve y=3x² and the lines x = -2, x = 3 and x-axis. Find the area under this curve.

Solution:

Curve is bounded by the lines x = −2 and x =3

Given Curve,

• y = 3x²

Area (A) under the curve and between the specified lines is given by the integral,

A = ∫_-2³ y.dx

A = ∫_-2³ 3x².dx

A = [x³]₋₂³ = 35

Practice Question on Application of Integrals

Q1. Determine the area within the circle defined by the equation x²+y²=r², where r represents the radius of the circle.

Q2. Find the area enclosed by the curve y=2x² and the horizontal line y=6.

Q3. Calculate the area bounded by the curve y²=4y−x and the y-axis.

Q4. Use integral calculus to find the area under the curve for the region enclosed by the ellipse x²/49 + y²/36 = 1.

Q5. Determine the region’s area under the curve y=2x² considering the lines x=−2, x=4, and the x-axis.

Application of Integrals-FAQs

1. What are Applications of Integrals?

Integrals play an important role in Mathematics, Science, and Engineering. In the terms of Physics, it is used in calculation of center of mass, moment of inertia, center of gravity, work-done, thrust, flow rate, volume, and pressure distribution. It is used in medical science to understand bacterial growth dynamics and food conditions. It is also used in understanding the principles of electromagnetism.

2. What Real-Life Applications of Integrals and Differentiation?

Applications of differentiation and integration are used to solve diverse types of real-world challenges. In derivatives, we identify the higher and lower points of any functions like cost, structural strength, material quantities in construction, profit, and loss. On the other hand, Integrals are used in determining the areas enclosed by irregular shapes. It offers a practical tool for addressing the spatial problems.

3. How are Integrals Applied in the Field of Mathematics?

In mathematics, Integrals are used in calculating area, volume, work, and analyzing the paths of functions in both Cartesian and polar coordinates.

4. What are Integrals?

An Integral is defined as a function that corresponds to the antiderivative of a given function. Its primary aim is to determine the areas of two-dimensional space and calculate volumes for three-dimensional objects. Integral is the representation of calculating the area under a curve with respect to x-axis which is equivalent to the integral of a function with respect to the x-axis. The process of finding out the anti-derivative is known as integration.

5. What are the Types of Integrals?

There are two type of Integrals.

• Indefinite Integrals: Indefinite integrals do not have predefined limits i.e. its upper and lower limits are not defined, resulting in an undetermined final value.
• Definite Integrals: Definite integrals are characterized by specific limits, representing the area under a curve within those fixed boundaries.

6. Can Definite Integrals have Negative Values?

Yes, definite Integrals have negative values. Definite Integrals are applied to measure the area between the x-axis and a given curve within a specific interval.

7. What are Practical Applications of Integrals?

Integrals find various applications in both scientific and mathematical contexts. It is used to determining the area under a curve, calculating the center of mass for objects, and addressing other mathematical and scientific problems.

8. What we have to study in Application of Integrals Class 12?

In Application of Intgerals Class 12 we have to study how to find area between curves using the integration method.