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 the general solution of the differential equation \[ y\log y\,\frac{dx}{dy}+x=\frac{2}{y}. \]
View Solution
2.
A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is \(3\) m as shown. Based on the given information answer the following:

(i) Express \(y\) as a function of \(x\) from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to \(x\).
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates of the points \(P\) and \(Q\) where the outer edge of the track cuts \(x\)-axis and \(y\)-axis in first quadrant and find the area of triangle formed by points \(P,O,Q\).
View Solution
3.
Mother, Father and Son line up at random for a family picture. Let events \(E\): Son on one end and \(F\): Father in the middle. Find \(P(E/F)\).
View Solution
4.
A line passing through the points \(A(1,2,3)\) and \(B(6,8,11)\) intersects the line \[ \vec r = 4\hat i + \hat j + \lambda(6\hat i + 2\hat j + \hat k) \] Find the coordinates of the point of intersection. Hence write the equation of a line passing through the point of intersection and perpendicular to both the lines.
View Solution
5.
Evaluate : \[ \int_{\frac{1}{12}}^{\frac{5}{12}} \frac{dx}{1+\sqrt{\cot x}} \]
View Solution
6.
Find the domain of \(p(x)=\sin^{-1}(1-2x^2)\). Hence, find the value of \(x\) for which \(p(x)=\frac{\pi}{6}\). Also, write the range of \(2p(x)+\frac{\pi}{2}\).
View Solution

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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 the general solution of the differential equation \[ y\log y\,\frac{dx}{dy}+x=\frac{2}{y}. \]

3.
Mother, Father and Son line up at random for a family picture. Let events \(E\): Son on one end and \(F\): Father in the middle. Find \(P(E/F)\).

5.
Evaluate : \[ \int_{\frac{1}{12}}^{\frac{5}{12}} \frac{dx}{1+\sqrt{\cot x}} \]

6.
Find the domain of \(p(x)=\sin^{-1}(1-2x^2)\). Hence, find the value of \(x\) for which \(p(x)=\frac{\pi}{6}\). Also, write the range of \(2p(x)+\frac{\pi}{2}\).

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 the general solution of the differential equation \[ y\log y\,\frac{dx}{dy}+x=\frac{2}{y}. \]

3.Mother, Father and Son line up at random for a family picture. Let events \(E\): Son on one end and \(F\): Father in the middle. Find \(P(E/F)\).

5.Evaluate : \[ \int_{\frac{1}{12}}^{\frac{5}{12}} \frac{dx}{1+\sqrt{\cot x}} \]

6.Find the domain of \(p(x)=\sin^{-1}(1-2x^2)\). Hence, find the value of \(x\) for which \(p(x)=\frac{\pi}{6}\). Also, write the range of \(2p(x)+\frac{\pi}{2}\).

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Find the general solution of the differential equation \[ y\log y\,\frac{dx}{dy}+x=\frac{2}{y}. \]

3.
Mother, Father and Son line up at random for a family picture. Let events \(E\): Son on one end and \(F\): Father in the middle. Find \(P(E/F)\).

5.
Evaluate : \[ \int_{\frac{1}{12}}^{\frac{5}{12}} \frac{dx}{1+\sqrt{\cot x}} \]

6.
Find the domain of \(p(x)=\sin^{-1}(1-2x^2)\). Hence, find the value of \(x\) for which \(p(x)=\frac{\pi}{6}\). Also, write the range of \(2p(x)+\frac{\pi}{2}\).