Application of Integrals – Notes, Topics, Formulas, Equations, Books, Question & Answers

Application of Integrals

Edited By Team Careers360 | Updated on May 07, 2022 01:00 PM IST

Introduction:
Functions, differentiation, and integration are all often used terms in calculus. It may be used in a variety of domains, including engineering, science, and mathematics. We will learn how to utilise integrals to find areas under various scenarios in this chapter. Before we get started, it's important to understand that talking about the area under a curve only makes sense when you have a graph of that curve. Let's look at an x-function with the formula y = f(x). The region below the curve is clearly limitless, as seen in the graph! To put it another way, if we don't define the x values within which the area under the curve must be calculated, our result will be indeterminable. It can be simpler to express/plot a given function y = f(x) as x = f(x) (y). It's also possible that a function in the form of x = f(y) is easier to integrate than a function in the form of y = f(y) (x). In such instances, the area under a curve with respect to the y-axis would equal the one. Our result is indeterminable if we don't provide the x values within which the area under the curve must be calculated. The limiting values of x establish the bounds of the region available beneath the curve

JEE Main 2025: Sample Papers | Mock Tests | PYQs | Study Plan 100 Days

JEE Main 2025: Maths Formulas | Study Materials

JEE Main 2025: Syllabus | Preparation Guide | High Scoring Topics

List of topics according to NCERT and JEE Main/NEET syllabus:

Introduction to the application of integrals
Areas and Simple curves
The area of a region bounded by a curve and a line
The area between the two curves

Important concepts and Laws:

Calculation The area under the curve method is a technique for calculating the area under the curve.

1st step: make a rectangular strip with height/length = f(x₀) and width = dx.The rectangle can be considered centered at the value x = x₀, where dx is an infinitesimally tiny number that can be assumed to equal the difference in the x-coordinates on which the rectangle's sides are positioned.

2nd step: Move the strip beneath the curve, starting at the lower bound of x, x = a, and ending at the higher bound of x, x = b, altering the value of x0 at each position while keeping the same value of dx.

3rd step: A single rectangular strip's area dA = length x width ?? = ?(x₀) × ??. This is known as the Differential Area.

4th step: By adding the areas of all the rectangular strips, the total area A beneath the curve may be approximated.x₀=bx₀=adA. Now substitute the value of dA from the third step and we get.x₀=bx₀=a?(x₀) × ??

5th step: when dx is tending to zero, the summation may be transformed into an integral. Then there's;

A= ab?(?)??

We calculated the area under the curve y = f(x) using a Riemann Sum, which approximated an integral by using a finite sum; the number of rectangular strips was finite, but we transformed the sum to an integral by taking that number (the same as choosing dx tending 0 ). We may now go on to the formal definition of the area under curves.

Integrals Have a Wide Range of Applications in Engineering: Integrals have a wide range of applications in engineering. To calculate the quantity of material necessary in a curved surface in architecture. Take, for example, the construction of a dome. Integrals are used in electrical engineering to determine the length of a power cable that is needed for transmission between two power plants. Integrals are used in a variety of fields, including medicine. Integrals are used to calculate the growth of bacteria in the laboratory while controlling for factors such as temperature and diet.

NCERT Notes Subject wise link:

Importance of application of integral class 12

We are already familiar with the various methods of integration in Maths from prior sessions. Students will learn how to use integrals to calculate the area under basic curves, the area between lines, and arcs of standard forms of different curves such as circles, parabolas, and ellipses in chapter 2 of class 12 math. We'll also look at determining the area enclosed by these curves, as well as the combination of lines. The chapters are still relevant and important in 12th grade, and the ideas we learn here will help us understand the topics we'll be learning for forthcoming competitive examinations like JEE mains, JEE advanced, State Engineering Entrance Exams, and so on. Furthermore, because of its importance in the 12th board test, it is one of the most crucial chapters for people who desire to excel in their board examinations.

NCERT Solutions Subject wise link:

NCERT Exemplar Solutions Subject wise link:

Frequently Asked Questions (FAQs)

1. What is the Application of Integrals and How Do They Work?

Areas under simple curves, the area between lines and arcs of circles, parabolas, and ellipses are all calculated using integrals. The use of integration to get an area between a curve and the x-axis is demonstrated. The area between a curve and the x-axis, as well as the area between a curve and a second curve, maybe interpreted with very little alteration. In addition, equations may be used to determine distance, velocity, acceleration, volume, the average value of a function, work, centre of mass, kinetic energy, improper integrals, probability, arc length, and surface area.

2. What's the Difference Between Integration and Differentiation?

Integration and differentiation, like square and square root functions, are connected to one another. Let's say we take the number x and square it to obtain X². We can now recover the original x by using the square root function on X². Similarly, we receive a new integral when we integrate a continuous function F. If we apply differentiation to this new integral, we will end up with the original function F.

3. Is it feasible to have a negative area between two curves?

No, there will be no negative area between the two curves. This is due to the fact that the area between the two curves differs from the region beneath the curve. As a result, the area between the two curves must be positive at all times.

4. Is it feasible to have a negative integral?

It is true that a definite integral can be negative. The negative result will occur if all of the areas within the interval is below the x-axis but still above the curve.

5. What does the area under the curve mean?

A definite integral between two locations can be used to compute the area under a curve that exists between them. To get the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between a and b's limits.

6. What does it mean to have a polar curve?

A polar curve is a form that is created with the use of the polar coordinate system. They are distinguished by the presence of points at varying distances from the pole or origin.