An area or region could be greatly affected depending upon several external factors. Therefore, determining the area falls under simple curves in Application of Integrals. Earlier in class, students were introduced to a definite integral, usually being the limit of a sum.

Now Chapter 8, Application of Integrals, delves into talking about it, but in a more efficient and intuitive manner by helping find the area bounded by the curve y = f (x), x-axis and the coordinates of x = a and x = b. Herein, students will explore the “elementary area” and its integration as well. This means it helps students find the area bounded by a curve of any shape, size, or quadrant.

In a nutshell, you will learn the process usually involved behind finding the area confined by simple curves, ellipses, parabolas, their interrelations and several practical approaches in Application Of Integrals Class 12 Ncert Solutions. One out of many examples is the graphical representation you will become acquainted with while studying the professionally curated details of the topic. It is extremely important that you learn the subtopics well because they will help you score well.

When introducing the third subtopic of Application of Integrals, the first question students usually want to have an answer to is how can they conveniently find The Area Between Two Curves that are defined by functions. Now, to find the Area Between Two Curves, it is essential that you integrate the difference between the two of them.

This is how you typically find the area between two curves in Application Of Integrals Class 12 Ncert Solutions. The topic discusses various formulae and techniques that students can utilise to extract answers from several types of questions. Every problem can be treated as its particular subset, while this cannot. Apart from being one essential application of definite integrals, several of these problems help provide a concrete idea about the nature of two curves used as functions.

This subtopic, namely The Area Between Two Curves, will undeniably help students become familiar with the concept of curves, their point of meeting, and the ways to find the area held within the unbounded region. In this context, since there are two functions, each of them is supposed to be plotted as a curve in a Cartesian plane, usually with the help of a horizontal axis in Application Of Integrals Class 12.

1. Is Integrals an important chapter in Integrals?

One of the concepts that is most essential and central to calculus is the determination of integrals (Application Of Integrals Class 12). Integration is commonly utilized in every form of mathematics, engineering and niches of science. Given its importance, Integrals undeniably have certain practical approaches which makes them a very significant part of NCERT Solutions (Application of Integrals) and require high attention. It helps in determining the area of an arbitrary shape, length of a curve or the typical volume of a solid, making the topic one very important part of maths that cannot be skipped. NCERT Solutions has curated and designed easy-to-grasp concepts so as to support interactive learning.

2. What exactly is an integral?

When a function is either integrated or differentiated, it generally expresses the area formed under the curve on a graph of the specified function. Integrals, in Application of Integrals, can be interpreted as the most elemental object in Calculus that helps summing infinitesimal pieces to find the region of a continuous, bounded path. It can simply be a curve adjoined with another or a continuous bounded object, like the length of a plotted graphical curve. It can help in finding the area or generalizing the area of a particular arbitrary shape, volume of a confined length. It can range anywhere from the area found in simple curves to The Area Between Two Curves. Integrals, in simple words, are the space usually seen under a graph of a given equation. It is important that you become acquainted with each form or shape, as found in Application Of Integrals Class 12.

3. Are integrals calculus?

Since integrals are part of the mathematical domain, they can be denoted as an area or the generalization of an area bounded by a region. Integrals are part of calculus and can generally help determine the area, volume, or length by combining infinitesimal data. Integrals can be majorly enumerated as certain definite integrals that can be defined as the area of a region confined by a graph of a specified function practically between two points. It would be wrong to assert that integrals are what calculus is because they are one part of it, exactly how engineering is one major part of science. That being said, integrals are a branch of the field of calculus, which is equally important to ace. 

4. What are the types of integrals?

Fundamentally, there are only two types of integrals in Application Of Integrals Class 12, namely,

• Definite Integrals: Definite Integrals essentially include a function that comes with limits of Integration. Associating two primary values as the limits for the interval of integration, definite integrals are known to be twofold. They are namely the lower limit and the upper limit. And, it also lacks any constant of integration.
• Indefinite Integral: Indefinite Integrals come into action when there has not been any limit for integration. And since it does not have any limit, it formulates an arbitrary constant for the same purposes. It is defined without the usage of lower or upper limits. 

5. What is the integral formula?

Integral formulae can be more than one. Now, the process typically used in finding the antiderivative of a function or a variable is known as integration. Since they are as important as the differentiation formulae, it requires your attention to detail. Here is a list of all the integral formulae,

• ∫ a dx = ax + C
• ∫ cos x dx = sin x + C
• ∫ sec2x dx = tan x + C
• ∫ csc2x dx = -cot x + C
• ∫ 1 dx = x + C
• ∫ sec x (tan x) dx = sec x + C
• ∫ csc x ( cot x) dx = – csc x + C
• ∫ sin x dx = – cos x + C
• ∫ (1/x) dx = ln |x| + C
• ∫ ex dx = ex+ C
• ∫ xn dx = ((xn+1) / (n+1))+C ; n≠1
• ∫ ax dx = (ax/ln a) + C ; a>0, a≠1

This set of formulae can be further classified as:

• Exponential functions
• Logarithmic functions
• Trigonometric functions
• Inverse trigonometric functions
• Hyperbolic functions
• Rational functions
• Irrational functions
• Inverse hyperbolic functions
• Gaussian functions

All of these are important to ace. You will learn the various purposes, properties and usages in Application of Integrals.