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NCERT Solutions for Miscellaneous Exercise Chapter 8 Class 12 - Application of Integrals

NCERT Solutions for Miscellaneous Exercise Chapter 8 Class 12 - Application of Integrals

Edited By Komal Miglani | Updated on May 08, 2025 02:31 PM IST | #CBSE Class 12th

Integrals are an inseparable part of calculus, which can solve real-world problems related to areas and volumes by summing up infinitely many small pieces to make a whole. The application of integrals delves into the aspect of how integrals can be used to solve problems related to real-life scenarios. The miscellaneous exercise of the chapter, Application of Integrals, combines all the key concepts covered in the chapter, so that the students can enhance their understanding by a comprehensive review of the entire chapter and get better at problem-solving. This article on the NCERT Solutions for Miscellaneous Exercise of Class 12, Chapter 8 - Application of Integrals, offers detailed and easy-to-understand solutions for the exercise problems, so that students can strengthen their understanding of the application of integrals. For syllabus, notes, exemplar solutions and PDF, refer to this link: NCERT.

This Story also Contains
  1. Application of Integrals Class 12 Chapter 8 Miscellaneous: Exercise
  2. Topics covered in Chapter 8, Application of Integrals: Miscellaneous Exercise
  3. NCERT Solutions Subject Wise
  4. NCERT Exemplar Solutions Subject Wise
LiveCBSE Board Result 2025 LIVE: Class 10th, 12th results soon on cbse.gov.in; marksheets, DigiLocker codeMay 8, 2025 | 2:28 PM IST

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Application of Integrals Class 12 Chapter 8 Miscellaneous: Exercise

Question 1: Find the area under the given curves and given lines:

(i) y=x2,x=1,x=2 and x -axis

Answer:

The area bounded by the curve y=x2,x=1,x=2 and x -axis
1594728126741
The area of the required region = area of ABCD
=∫12ydx=∫12x2dx=[x33]12=73
Hence the area of shaded region is 7/3 units

Question 1: Find the area under the given curves and given lines:

(ii) y=x4,x=1,x=5 and x -axis

Answer:

The area bounded by the curev y=x4,x=1,x=5 and x -axis

1594728286834
The area of the required region = area of ABCD
=∫15ydx=∫12x4dx=[x55]12=625−15=624.8
Hence the area of the shaded region is 624.8 units


Question 2: Sketch the graph of y=|x+3| and evaluate ∫−60|x+3|dx.

Answer:

y=|x+3|

the given modulus function can be written as

x+3>0

x>-3

for x>-3

y=|x+3|=x+3

x+3<0

x<-3

For x<-3

y=|x+3|=-(x+3)

1654760706138

Integral to be evaluated is

∫−60|x+3|dx=∫−6−3(−x−3)dx+∫−30(x+3)dx=[−x22−3x]−6−3+[x22+3x]−30=(−92+9)−(−18+18)+0−(92−9)=9

Question 3: Find the area bounded by the curve y=sin⁡x between x=0 and x=2π .

Answer:

The graph of y=sinx is as follows

1654760755958

We need to find the area of the shaded region

ar(OAB)+ar(BCD)

=2ar(OAB)

=2×∫0πsinxdx=2×[−cosx]0π=2×[−(−1)−(−1)]=4

The bounded area is 4 units.




Question 4: Choose the correct answer.

Area bounded by the curve y=x3 , the x -axis and the ordinates x=−2 and x=1 is

(A) −9 (B) −154 (C) 154 (D) 174

Answer:

1654765098486

Hence the required area

=∫−21ydx

=∫−21x3dx=[x44]−21

=[x44]−20+[x44]01

=[0−(−2)44]+[14−0]

=−4+14=−154

Therefore the correct answer is B.

Question 5: Choose the correct answer.

T he area bounded by the curve y=x|x| , x -axis and the ordinates x=−1 and x=1 is given by

(A) 0 (B) 13 (C) 23 (D) 43

[ Hint : y=x2 if x>0 and y=−x2 if x<0 . ]

Answer:

The required area is

2∫01x2dx=2[x33]01=23 units


Also Read,

Topics covered in Chapter 8, Application of Integrals: Miscellaneous Exercise

The main topics covered in class 12 maths chapter 8 of Application of Integrals, Miscellaneous Exercise are:

  • Area under curves: In this topic, we will calculate the area between a curve and the coordinate axes in a specific interval. For example, the area under the curve y=f(x), between two points on the X axis, as x=a and x=b, can be found using definite integrals as: A=∫abf(x)dx.
  • Area between two curves: This topic deals with the area between two curves. Let f(x) and g(x) be two curves in the interval [a,b], then the area can be found using the formula, Area =∫ab[f(x)−g(x)]dx.
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NCERT Solutions Subject Wise

Below are some useful links for subject-wise NCERT solutions for class 12.

JEE Main Important Mathematics Formulas

As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters

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NCERT Exemplar Solutions Subject Wise

Here are some links to subject-wise solutions for the NCERT exemplar class 12.

Frequently Asked Questions (FAQs)

1. How many questions are there in Miscellaneous exercise Chapter 8 ?

There are 19 questions total in Miscellaneous exercise Chapter 8.

2. Can we find an area without using integrals ?

Simple figures like triangle, circle etc. can be tackled without integration but not the complex ones. 

3. Are questions repeated in the examination from this Chapter ?

Yes, in the Board exam the questions are repeated every year. 

4. What is the level of questions asked from this Chapter ?

Moderate level questions are asked from this Chapter. 

5. Can one skip Miscellaneous exercise ?

No, as it has some good questions, miscellaneous exercise must be done. 

6. What is the time it will take to complete for the first time ?

It will take around 5-6 hours to complete for the first time.

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Manasvini Gupta 30 January,2025
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Kanishka kaushikii 17 September,2024
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Ashvadha 12 July,2024
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