NCERT Solutions for Class 12 Maths Chapter-8 Application of integrals

NCERT Solutions for Class 12 Maths Chapter-8 Application of integrals

In chapter 7 Integrals, we already study that to find the area bounded by the curve $\dpi{100} y = f (x)$, $\dpi{100} x = a, x = b$ and $\dpi{100} x-axis$, while calculating the definite integral as the limit of a sum. In Chapter-8 Application of , we will learn some specific applications of integrals to find the area under curves, the area between the lines and arcs of circles, ellipses, and parabolas. We will also learn how to find the area bounded by the above-said curves. In this chapter, there are 2 exercises with 20 questions. The  NCERT Solutions for Class 12 Maths Chapter-8 Application of integrals are solved by our subject experts to help students in their CBSE board exams and in competitive exams like JEE Main.

For the students to understand  Chapter 8 Application of integrals, in a better way total 14 solved examples are given and to develop a grip on the topic, at the end of the chapter, 19 questions are given in a miscellaneous exercise. Important topics that we are going to learn in NCERT  Class 12 Maths Chapter-8 Application of integrals are- find the area under simple Curves, find the area of the region bounded by a curve and a line and find the area between two curves.

• How to find the area under Simple Curves-To find the area bounded by the curve $\dpi{100} y= f(x), \:x-axis$  and the ordinates $\dpi{100} x = a$ and $\dpi{100} x = b$. Let assume that area under the curve as composed of large numbers of very thin vertical strips. Consider an arbitrary strip of width dx and height y ,then area of the elementary strip(dA) = ydx, where, y = f(x). This small area called the elementary area.

$\dpi{100} \\A=\int_{a}^{b}dA\\A=\int_{a}^{b}ydx\\A=\int_{a}^{b}f(x)dx$

• How to find the area of the region bounded by a curve and a line- In this subsection, we will find the area of the region bounded by a line and a circle, a line and an ellipse, a line and a parabola. Equations of above-said curves will be in their standard forms only. Let's understand with an example-

Example- Find the area of the region bounded by the curve $\dpi{100} y=x^2$ and the line y = 4.

Solution- Since the given curve represented by the equation $\dpi{100} y=x^2$ is a parabola symmetrical about y-axis only, therefore, from Fig 8.9, the required area of the region AOBA is given by

$\dpi{100} \\=2\int_{0}^{4}xdy\\=2\left ( area \:of \:the \:region \:BONB \:\:bounded \:by \:curve \:y-axis \:and \:\:the \:lines\:y=0 \:and\:y=4 \right )\\=2\int_{0}^{4}\sqrt{y}dy\\=2\times \frac{2}{3}\left [ y^{\frac{3}{2}} \right ]_{0}^{4}\\=\frac{32}{3}$

Topics and sub-topics of NCERT Grade 12 Maths Chapter-8 Application of integrals-

8.1 Introduction

8.2 Area under Simple Curves

8.2.1 The area of the region bounded by a curve and a line

8.3 Area between Two Curves

NCERT Solutions for Class 12 Maths Chapter-8 Application of integrals- Solved Exercise Questions

NCERT Solutions for Class 12 Maths Chapter 8 Application of integrals Exercise 8.1

NCERT Solutions for Class 12 Maths Chapter 8 Application of integrals Exercise 8.2

NCERT Solutions for Class 12 Maths Chapter 8 Application of integrals Miscellaneous

NCERT Solutions for class 12- Maths

 Chapter 1 Relations and Functions Chapter 2 Inverse Trigonometric Functions Chapter 3 Matrices Chapter 4 Determinants Chapter 5 Continuity and Differentiability Chapter 6 Application of Derivatives Chapter 7 Integrals Chapter 9 Differential Equations Chapter 10 Vector Algebra Chapter 11 Three Dimensional Geometry Chapter 12 Linear Programming Chapter 13 Probability