# NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals

## NCERT Solutions for Class 12 Maths Chapter 8 - Application of Integrals You can now download NCERT Solutions Class 12 Maths Chapter 8 PDF available at the official website of Vedantu. Our Application of Integrals Class 12 NCERT Solutions is made in an easily understandable manner by the subject matter experts of our team. The teachers are well-versed with the current CBSE syllabus, and their solutions are designed based on that. With these NCERT Solutions for Class 12 Maths Chapter 8, you would be able to quickly revise all the salient points of the chapter and remember all the important formulas.

Ch 8 Maths Class 12 is based on the application of integrals which can be a tricky topic for many of you. In order to master it, you need support from experts who have a thorough knowledge of this topic, and that’s what our team is well equipped with. On top of the ease of access to quality NCERT Solutions for Class 12 Maths Chapter Application of Integrals on our website, you also get constant support from our subject matter experts in case you get any problem on any topic.

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Having easy access to Maths solutions can give the required boost needed by students in today’s hectic lifestyle. With the readily available NCERT Solutions Class 12 Maths Chapter 8 PDF download at our website, students can revise their syllabus on the go.

### 8.1 Introduction

In the introduction of Chapter 8 Class 12th Maths, you will be reminded of all the geometrical formulae you learned in the previous chapters. You would recall how these geometric formulae for calculating areas of triangles, rectangles, trapeze, etc. are the basis on which mathematics is applied to real-life problems. However, they are not able to determine areas enclosed by curves. That is where integral calculus comes into the picture.

You will rekindle what you learned about areas bounded by a curve in the previous chapter where definite integrals were calculated as the limit of a sum. In this chapter, you would use those concepts to find out areas between lines and arcs of circles, parabolas, simple curves and ellipses.

AOI Class 12 NCERT Solutions have many solved examples and a variety of questions neatly divided into different exercises. You will get rigorous practice with the set of questions presented in this chapter. Some of the prominent parts discussed here are:

• Elementary area i.e. the area located between any arbitrary positions

• The area under two curves

• The area that is bounded by a curve and a line

• Integrals of trigonometric identities

### 8.2 Area Under Simple Curves

This section of Chapter 8 Class 12 Maths begins with a recap of how definite integrals are calculated as the limit of a sum and the fundamental theorem of calculus. You will then learn how to find the area bounded by the curves. So, if the curve is y = f(x) then to find the area bounded by the curve, the x-axis, and 2 points x=p1 and x=p2 on the x-axis you can consider it as a large number of vertical strips. After this, you would understand how to calculate individual areas of these thin strips. Suppose the height of the strip is y and width is dx then the area of one strip is expressed as dA = ydx, here y = f(x). This area is termed as an elementary strip. The total area can thus be calculated by integrating these elementary strips:

A = $\int_{p1}^{p2}$ dA = $\int_{p1}^{p2}$ ydx = $\int_{p1}^{p2}$ f(x) dx

In case the position of the curve is below the x-axis, the area will come out to be negative, but we take only the absolute value of the integral.

You may learn how to calculate the area of a region which is bounded by a curve and a line. One could consider either vertical or horizontal strips to calculate the area of the region. So if the curve is y = f(x) and the equation of the line is g = ax + k, we can see there are 2 cases; one is the area under the curve, and the other is the area between 2 curves, depending on how many points the line intersects the curve at. For a line intersecting the curve at 2 points, a general formula to find the region between the line and the curve can be given as:

A = $\int_{p1}^{p2}$ [g(x) - f(x)]dx. Here g(x) is the equation of the line, f(x) is the equation of the curve, and p1, p2 are points of intersection of the curve with the straight line.

### 8.2 The Area Between Two Curves

In this unit of Application of Integrals Class 12 chapter, you would learn how to calculate area between 2 curves by dividing the common region into elementary areas of vertical strips. So if equation of one curve is y = f(x) and equation of the other curve is y = g(x) and we know that f(x) >= g(x), the elementary area can be given as: dA = [f(x) – g(x)]dx where dx is the width of the strip. The total area is thus calculated by the following integral:

A = $\int_{p1}^{p2}$ [f(x) - g(x)]dx. Here p1 and p2 are the 2 points of intersections of the curve on the x-axis.

### Key Features of NCERT Solutions for Class 12 Maths Chapter 8

To score well in CBSE and also crack many competitive exams like NEET and IIT, one needs to have a solid base in Mathematics. The NCERT books present you with challenging questions which better your analytical abilities and give you ample exposure to all kinds of questions that can come in any of these exams. Our Application of Integrals Class 12 Solutions is in line with the latest CBSE curriculum and is extremely beneficial for your preparation for the following reasons:

• The quality of solutions is impeccable as they are designed by teachers with immense experience in the subject matter.

• You get to revise all the key points of a chapter in very less time.

1. Which chapters are removed from maths Class 12?

According to the revised CBSE syllabus for Class 12 Maths, the syllabus has been reduced by 30%. Full chapters have not been deleted but certain portions from each unit have been removed from the CBSE Class 12 Maths syllabus. To get full NCERT Solutions Maths Chapter 8 based on the latest syllabus, download the NCERT Solutions Maths Chapter 8 PDF available only on Vedantu. Extra important questions are also solved for a better understanding.

2. How many important examples are there in Class 12 Maths Chapter 8 Miscellaneous Exercise?

There are five examples (from Example 11 to Example 15) given in the Miscellaneous Examples section in Class 12 Maths Chapter 8. These miscellaneous examples can be used to solve the Class 12 Maths Chapter 8 Miscellaneous Exercise given in the NCERT textbook. For fully solved exercise questions including the miscellaneous exercise present in Chapter 8, download the NCERT Solutions Class 12 Maths Chapter 8 PDF from Vedantu today and start practising!

3. How can I understand the chapters in Class 12 Maths?

NCERT Class 12 Maths can be intimidating at first, but if you strategize your study, you can overcome your fear of Class 12 Maths easily. The best way to understand all the chapters in Class 12 Maths is to refer to NCERT Solutions Class 12 Maths on Vedantu. All the solutions in Class 12 Maths on Vedantu have full explanations for each topic and sub-topics including exercise questions.

4. What is the underlying concept of Chapter 8 Application of Integrals?

The underlying concept of NCERT Class 12 Maths Chapter 8 Application of Integrals deals with the usage of geometric formulae for calculating the area of different geometric shapes and the area between two curves, area between curve and line, area between arbitrary positions, and trigonometric identities integrals. To get solutions to all exercises, you must refer to NCERT Class 12 Maths Chapter 8 on Vedantu.

5. What are the most important formulas that you need to learn in Chapter 8 Class 12 Maths?

The most important formulae that you need to learn in Class 12 Maths Chapter 8 are - formula for the area under the simple curve (area enclosed by the given circle, area enclosed within ellipse), the formula for the area of the region bounded by line and curve, and formula for the area between two curves. You can get access to all the solutions if you download the NCERT Solutions Class 12 Maths Chapter 8 on Vedantu at free of cost. These solutions are also available on the Vedantu app. SHARE TWEET SHARE SUBSCRIBE