# CBSE - Class 12 - Mathematics - Application of Integrals - NCERT Solutions

## NCERT Solutions for Mathematics Application of Integrals

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**NCERT solutions for class 12 Mathematics**

**NCERT 12 Mathematics Text book Solutions**

NCERT 12th class Mathematics book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Mathematics chapter wise NCERT solution for Mathematics part 1 and Mathematics part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

**NCERT Class 12 Mathematics Chapter-wise Solutions**

**Part I**

- Chapter 1: Relations and Functions
- Chapter 2: Inverse Trigonometric Functions
- Chapter 3: Matrices
- Chapter 4: Determinants
- Chapter 5: Continuity and Differentiability

**Part II**

- Chapter 6: Application of Derivatives
- Chapter 7: Integrals
- Chapter 8: Application of Integrals
- Chapter 9: Differential Equations
- Chapter 10: Vector Algebra
- Chapter 11: Three-Dimensional Geometry
- Chapter 12: Linear Programming
- Chapter 13: Probability

**CBSE class 12th Mathematics have two books. Each book has chapters and topics.**

**NCERT Mathematics Book Class 12 Part-1****NCERT Mathematics Book Class 12 Part-2**

Here is the list of topics covered under each chapter of class 12 Mathematics NCERT text book.

**1. Relations and Functions**

- 1.1 Introduction
- 1.2 Types of Relations
- 1.3 Types of Functions
- 1.4 Composition of Functions and Invertible Function
- 1.5 Binary Operations

**2. Inverse Trigonometric Functions**

- 2.1 Introduction
- 2.2 Basic Concepts
- 2.3 Properties of Inverse Trigonometric Functions

**3. Matrices**

- 3.1 Introduction
- 3.2 Matrix
- 3.3 Types of Matrices
- 3.4 Operations on Matrices
- 3.5 Transpose of a Matrix
- 3.6 Symmetric and Skew Symmetric Matrices
- 3.7 Elementary Operation (Transformation) of a Matrix
- 3.8 Invertible Matrices

**4. Determinants**

- 4.1 Introduction
- 4.2 Determinant
- 4.3 Properties of Determinants
- 4.4 Area of a Triangle
- 4.5 Minors and Cofactors
- 4.6 Adjoint and Inverse of a Matrix
- 4.7 Applications of Determinants and Matrices

**5. Continuity and Differentiability**

- 5.1 Introduction
- 5.2 Continuity
- 5.3 Differentiability
- 5.4 Exponential and Logarithmic Functions
- 5.5 Logarithmic Differentiation
- 5.6 Derivatives of Functions in Parametric Forms
- 5.7 Second Order Derivative
- 5.8 Mean Value Theorem

**6. Application of Derivatives**

- 6.1 Introduction
- 6.2 Rate of Change of Quantities
- 6.3 Increasing and Decreasing Functions
- 6.4 Tangents and Normal
- 6.5 Approximations
- 6.6 Maxima and Minima

**7. Integrals**

- 7.1 Introduction
- 7.2 Integration as an Inverse Process of Differentiation
- 7.3 Methods of Integration
- 7.4 Integrals of some Particular Functions
- 7.5 Integration by Partial Fractions
- 7.6 Integration by Parts
- 7.7 Definite Integral
- 7.8 Fundamental Theorem of Calculus
- 7.9 Evaluation of Definite Integrals by Substitution
- 7.10 Some Properties of Definite Integrals

**8. Application of Integrals**

- 8.1 Introduction
- 8.2 Area under Simple Curves
- 8.3 Area between Two Curves

**9. Differential Equations**

- 9.1 Introduction
- 9.2 Basic Concepts
- 9.3 General and Particular Solutions of a Differential Equation
- 9.4 Formation of a Differential Equation whose General Solution is given
- 9.5 Methods of Solving First order, First Degree Differential Equations

**10. Vector Algebra**

- 10.1 Introduction
- 10.2 Some Basic Concepts
- 10.3 Types of Vectors
- 10.4 Addition of Vectors
- 10.5 Multiplication of a Vector by a Scalar
- 10.6 Product of Two Vectors

**11. Three Dimensional Geometry**

- 11.1 Introduction
- 11.2 Direction Cosines and Direction Ratios of a Line
- 11.3 Equation of a Line in Space
- 11.4 Angle between Two Lines
- 11.5 Shortest Distance between Two Lines
- 11.6 Plane
- 11.7 Coplanarity of Two Lines
- 11.8 Angle between Two Planes
- 11.9 Distance of a Point from a Plane
- 11.10 Angle between a Line and a Plane

**12. Linear Programming**

- 12.1 Introduction
- 12.2 Linear Programming Problem and its Mathematical Formulation
- 12.3 Different Types of Linear Programming Problems

**13. Probability**

- 13.1 Introduction
- 13.2 Conditional Probability
- 13.3 Multiplication Theorem on Probability
- 13.4 Independent Events
- 13.5 Bayes' Theorem
- 13.6 Random Variables and its Probability Distributions
- 13.7 Bernoulli Trials and Binomial Distribution

**NCERT Solution Chapter 8: Application of Integrals**

In geometry, we have learnt formulae to calculate areas of various geometrical figures including triangles, rectangles, trapezias and circles. Such formulae are fundamental in the applications of mathematics to many real life problems. The formulae of elementary geometry allow us to calculate areas of many simple figures. However, they are inadequate for calculating the areas enclosed by curves. For that we shall need some concepts of Integral Calculus.

In the previous chapter, we have studied to find the area bounded by the curve y = f (x), the ordinates x = a, x = b and x-axis, while calculating definite integral as the limit of a sum. Here, in this chapter, we shall study a specific application of integrals to find the area under simple curves, area between lines and arcs of circles, parabolas and ellipses (standard forms only). We shall also deal with finding the area bounded by the above said curves.

**NCERT Solutions for Class 12th Mathematics**

NCERT Solutions Class 12 Mathematics PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Mathematics includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Mathematics have total 13 chapters. Class 12 Mathematics ncert Solutions in pdf for free Download are given in this website. Ncert Mathematics class 12 solutions PDF and Mathematics ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide.

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