NCERT Solutions For Class 12 Mathematics Chapter 8: Applications of the Integrals

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NCERT Solutions for class 12 mathematics Chapter 8 Applications of the Integrals cover important concepts of Area Between Two Curves, lines, parabolas; area of circles/ellipses. Application of Integrals covers the basic properties of integrals as well as the fundamental theorem of calculus. Applications of the Integrals will help students learn to find a function when its derivative is given and will also learn to find the area under a graph of a function.

Download: NCERT Solutions for Class 12 Mathematics Chapter 8 pdf

Class 12 Maths NCERT Solutions Chapter 8 Applications of the Integrals

Important Topics in Class 12 Mathematics Chapter 8 Applications of Integrals

Importabt concepts of Class 12 Maths covered in Chapter 8 Application Of Integrals of NCERT Solutions are:

Introduction to Applications of Integrals

The introduction section of this topic includes recollection of the idea of finding areas bounded by the curve. Definite integral as the limit of a sum, introduces different applications of integrals like the area under simple curves, between lines, parabolas and ellipses.

Average value of a function can be calculated using integration.

Example: Derivative of f(x) = x³ is f’(x) = 3x2; and the antiderivative of g(x) = 3x2 is f(x) = x³. Here, the integral of g(x) = 3x2 is f(x)=x³

Area Under Simple Curves

This section defines the area bounded by a curve. Area Under a Simple Curve is expressed using formula: y = f(x).

Area Between Two Curves

Area Between Two Curves section covers the method of finding the area between two curves with solved problems. Area can be found by dividing a certain region into a number of pieces of small area and then adding up the area of those tiny pieces. It is easier to find the area if the tiny pieces are vertical in shape.

Important Concepts of Area Between Two Curves:

Area of the region bounded by the curve y = f (x), x-axis and the lines x = a and x = b (b > a) is given by the formula: Area = \(\oint_a^b y dx=\oint_b^a f(x) dx\)
Area of the region enclosed between two curves y = f (x), y = g (x) and the lines x = a, x = b is given by the formula, Area = \(\oint_a^b\); where f(x) ≥ g(x) in [a, b]
If f (x) ≥ g (x) in [a, c] and f (x) ≤ g (x) in [c, b], a < c < b, then Area = \(\oint_a^c + \oint_c^b\)

NCERT Solutions For Class 12 Maths Chapter 8 Exercises

The detailed solutions for all the NCERT Solutions for Chapter 8 Applications of Integrals under different exercises are as follows:

Exercise 8.1 Solutions: 13 Questions (8 Long, 3 Short, 2 MCQs)
Exercise 8.2 Solutions: 7 Questions(5 Long, 2 MCQs)
Miscellaneous Exercise Solutions: 19 Questions (8 Long, 7 Short, 4 MCQs)

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CBSE CLASS XII Related Questions

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).
- \(p = -\frac{2}{3}, \, q = \frac{5}{3}\)
- \(p = -\frac{8}{3}, \, q = \frac{20}{3}\)
- \(p = \frac{20}{3}, \, q = -\frac{8}{3}\)
- \(p = 0, \, q = 0\)
View Solution
2.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]
View Solution
3.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :
4.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]
View Solution
5.
An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box i is selected and a cap picked out is i/6, where i = 1, 2, 3.
Based on the above information, answer the following questions :
6.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).
- \(\begin{bmatrix} \frac{1}{3} & 3 \\ 0 & \frac{4}{3} \end{bmatrix}\)
- \(\begin{bmatrix} \frac{1}{3} & 0 \\ 3 & \frac{4}{3} \end{bmatrix}\)
- \(\begin{bmatrix} \frac{4}{3} & 3 \\ 1 & 0 \end{bmatrix}\)
- \(\begin{bmatrix} \frac{4}{3} & 0 1 & \frac{3}{3} \end{bmatrix}\)
View Solution

View All

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Mathematics NCERT Solutions

NCERT Solutions For Class 12 Mathematics Chapter 13: Probability

NCERT Solutions For Class 12 Mathematics Chapter 9: Differential Equations

NCERT Solutions For Class 12 Mathematics Chapter 7 Integrals

NCERT Solutions For Class 12 Mathematics Chapter 12: Linear Programming

NCERT Solutions for Class 12 Maths Chapter 3 Matrices

NCERT Solutions For Class 12 Mathematics Chapter 6 Applications of Derivatives

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You can also check

NCERT Solutions For Class 12 Mathematics Chapter 8: Applications of the Integrals

Class 12 Maths NCERT Solutions Chapter 8 Applications of the Integrals

Important Topics in Class 12 Mathematics Chapter 8 Applications of Integrals

NCERT Solutions For Class 12 Maths Chapter 8 Exercises

CBSE CLASS XII Related Questions

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

4.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

6.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

Similar Mathematics Concepts

Comments

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NCERT Solutions For Class 12 Mathematics Chapter 8: Applications of the Integrals

Class 12 Maths NCERT Solutions Chapter 8 Applications of the Integrals

Important Topics in Class 12 Mathematics Chapter 8 Applications of Integrals

NCERT Solutions For Class 12 Maths Chapter 8 Exercises

CBSE CLASS XII Related Questions

1. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3.At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.On the basis of the above information, answer the following questions :

4. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

6.Find:Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

4.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

6.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).