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Important Questions for CBSE Class 12 Maths Chapter 8 – Application of Integrals
Integral Calculus is an essential part of maths. It is mainly used to calculate the areas enclosed by curves. Important Questions Class 12 Maths Chapter 8 will help students revise the chapter and practise more questions. Only the important and relevant questions have been included here. Students can take a look at these questions while preparing for the board exam. These questions follow the CBSE pattern. So, students can rely on them completely and practise more to score well. This chapter is very important from the exam perspective.
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You can also find CBSE Class 12 Maths Important Questions ChapterbyChapter Important Questions here:
CBSE Class 12 Maths Important Questions  
Sr No  Chapter No  Chapter Name 
1  Chapter 1  Relations and Functions 
2  Chapter 2  Inverse Trigonometric Functions 
3  Chapter 3  Matrices 
4  Chapter 4  Determinants 
5  Chapter 5  Continuity and Differentiability 
6  Chapter 6  Application of Derivatives 
7  Chapter 7  Integrals 
8  Chapter 8  Application of Integrals 
9  Chapter 9  Differential Equations 
10  Chapter 10  Vector Algebra 
11  Chapter 11  Three Dimensional Geometry 
12  Chapter 12  Linear Programming 
13  Chapter 13  Probability 
CBSE Class 12 Maths Chapter8 Important Questions
Long Answer Type Questions (6 Marks)
Students can view the sample questions for long answers provided here. For the complete set of questions, they can click the link given.
Q.1. Find the area of the circle x2 + y2 = a2 .
Ans. Draw the circle.
Centre of the circle = (0, 0); radius = a.
Therefore, OA = OB = a = radius of the circle, where, A = (a, 0); B = (0, a).
Since the area of the circle is symmetric about the X axis and the Y axis.
Therefore, the area of the circle = 4 X area of OBAO
= 4 X 0aydx
Again, x2 + y2 = a2
y2 = a2 – x2
y = ±a2–x2
The positive value of y will be considered as AOBA lies in the 1st quadrant.
So, y = a2–x2
Now the area of the circle = 4 X 0aa2–x2 dx.
We know that, a2–x2 dx = 12a2–x2 + a22 xa + c.
= 4 X x2 a2–x2+ a22 xa a 0
= 4 x a2 a2–a2+ a22 aa – 02 a2–02+ 022 0
= 4 X 0+ a22 (1) 00
= 4 X a22 (1)
= 4 X a22 X 2
= πa2
Q.2. Find the common area bounded by circles x2 + y2 = 4 and (x – – 2)2 + y2 = 4.
Ans. The equations of the given circles are
x2 + y2 = 4 ………… (1)
(x – – 2)2 + y2 = 4 …………. (2)
The position of the centre of the first circle is at the origin (0, 0) and the radius is 2 units, from equation (1).
From equation (2), we get that the centre is at (2, 0) and on the x axis; the radius is 2 units.
Therefore, by solving the equations we get the intersecting points of the circles P = 1, 3, and Q = 1, 3.
Since one of the two circles is symmetric about the x axis,
Therefore, the required area
= 2 (area OPACO)
= 2 (area OPCO + area CPAC)
= 2 [area OPCO (part of the circle (x 2)2 + y2 = 4) + area CPAC (part of the circle x2 + y2 = 4)
= 2 y dx ( for circle (x 2)2 + y2 = 4) + y dx (for the circle x2 + y2 = 4)
= 2 014x22dx+ 124x2dx
= 2 x22 4x22 + 42 (x2)2 1 0 +x2 4 x2+ 42 x2 2 1
= 2 122 4 122+ 42 122 – 22 44 + 42 22 + 32 43+ 42 12
= 2 –12 4 1+ 42 12 + 44 – 42 1+ 42 1 –32– 42 12
= 2 –12 × 3 – 42 12 +42 1+ 42 1 32 – 42 × 6
= 2 – 3– 3+π+π 3
= 2 – 3– 2π3+2π
= 2 6π2π3– √3
= 2 4π3– √3
= 8π32√3 sq. unit.
3. Using integration, find the area of the region bounded by the triangle whose vertices are (–2, 2) (0, 5) and (3, 2).
Ans. Let the triangle be ABC. The area of ∆ ABC is to be found. The vertices are A (2, 2), B (0, 5), and C (3,2).
The equation of AB is as follows:
y − y1 = y2 – y1x2– x1x x1
y – 2 = 520+2 x+2
y – 2 = 32 x+2
Y = 32x + 5 ……………………….. (i)
The equation of BC,
y – 5 = 2 530 x0
y – 5 = – 33 x0
y – 5 = − x
Y = 5 – x ………………………… (ii)
The equation of AC,
y – 2 = 223+2 x+2
y – 2 = 0
y = 2 ………………………….. (iii)
The required area (Z)
= {(Area between line AB and x axis) – (Area between line AC and x axis) from x = − 1 to x = 0} + {(Area between line BC and x axis) – (Area between line AC and x axis) from x = 0 to x= 3}
Let, {(Area between line AB and x axis) – (Area between line AC and x axis) from x = − 1 to x = 0} = Z1
And, {(Area between line BC and x axis) – (Area between line AC and x axis) from x = 0 to x= 3} = Z2
Therefore, required area Z = Z1 + Z2
Z1 = 203x2+5dx
= 203x2+52dx
= 203x2+3dx
= 3 x24+x0 2
= 3 0– 442
= 3
Z1 = 3
Z2 = 03y2– y3dx
= 035x2dx
= 033xdx
= 3x x223 0
= 9 92
= 92
So, the area bounded by the triangle is 3 + 92 = 152 sq. units.
4. Find the area enclosed by the ellipse x2a2+ y2b2=1.
Ans. We have to find the area enclosed by the ellipse. The given equation is, x2a2+ y2b2=1.
The ellipse is symmetrical in both the axes.
So, the area of the ellipse = 4 × aarea of OAB.
= 4 0ay dx
It is given that, x2a2+ y2b2=1
y2b2=1 − x2a2
y2b2 =a2– x2a2
y2 = b2a2a2– x2
y = b2a2a2– x2
y = ± ba a2– x2
The ellipse is in the 1st quadrant,
So, the value of y must be positive.
Therefore, y = ba a2– x2
Area of ellipse
= 4 × 0ay dx
= 4 0aba a2– x2 dx
= 4ba 0aa2– x2 dx
It is of form,
a2– x2 dx
= 12x a2– x2 + a22 xa + c
= 4ba x2 a2– x2+ a22 xa a 0
= 4ba a2 a2– a2+ a22 aa – 02 a20– a22 0
= 4ba 0+ a22 100
= 4ba × a22 1
= 2ab × 1
= 2ab × 2
= πab
Therefore, the area enclosed by the ellipse is πab sq. units.
5. Find the area of the region bounded by the x axis, the line y = x, and the circle x2 + y2 = 32. The region is in the first quadrant.
Ans. The area is bounded by,
The x axis ………………… (i)
The line y = x …………………… (ii)
And the circle x2 + y2 = 32 ……………………… (iii)
From the equation of the circle we get,
x2 + y2 = 32
x2 + y2 = 16 × 2
x2 + y2 = 42 2
x2 + y2 = 4√22
Let the line and the circle intersect at point M.
Therefore, the required area = area of OMA
First, we have to find the coordinates of M.
From equation (ii) we get that y =x. Putting the value of y in equation (iii) we get,
x2 + y2 = 32
x2 + x2 = 32
2x2 = 32
x2= 16
x = ±4
Therefore, x = +4, and x = −4
y = 4, when x = 4;
y = −4, when x = −4.
So, the possible interesting points are (4, 4), (−4, −4).
Since the region is in the first quadrant, the coordinates of M must be (4, 4).
Required area = Area OMP + Area PMA = 04y1 dx+ 44√2y2 dx
x2 + y2 = 4√22
y2 = 422 – x2
y = ± 422 – x2
Only the positive value of y will be considered as the region is in the first quadrant.
Therefore, y = 422 – x2
Required area = 04x dx+ 44√24√22– x2 dx
Let, I1 = 04x dx and I2 = 44√24√22– x2 dx
Solving the first part we get,
I1
= x224 0
= 4202
= 162
= 8
Solving the second part,
I2
= 44√24√22– x2 dx
It is in the form of a2– x2 dx = 12xa2– x2 + a22 xa + c
So, replace a by 4√2,
I2 = x2 (4√2)2– x2+ (42)22 x4√2 4√2 4
= 4√224√22–4√22 + 4√2224√24√2 − 424√22– 42 − 4√22244√2
= 0 + 16×221 − 23216 − 16×221√2
= 161 − 216161√2
= 16 1 – 1√2 − 8
= 16 2 – 4 − 8
= 16 4π2π4×2 – 8
= 1682π – 8
= 2[2] – 8
= 4π8
Therefore, I1 + I2
= 8 + 4π8
= 4
So, the required area = 4π sq. units.
Q1.
Find the area of the region bounded by x^{2} = y and y = x.
$\begin{array}{l}\\ \\ \end{array}$
Ans.
$\begin{array}{l}\mathrm{From}\mathrm{y}=\left\mathrm{x}\right\mathrm{and}{\mathrm{x}}^{2}=\mathrm{y},\mathrm{we}\mathrm{get}\\ \mathrm{x}=1,1\\ \mathrm{The}\mathrm{required}\mathrm{area}=2\mathrm{x}\left[\underset{0}{\overset{1}{\u02c6\xab}}{\mathrm{y}}_{\mathrm{L}}\mathrm{dx}\underset{0}{\overset{1}{\u02c6\xab}}{\mathrm{y}}_{\mathrm{p}}\mathrm{dx}\right]\\ =\mathrm{}2\mathrm{x}\left[\underset{0}{\overset{1}{\xab}}\mathrm{xdx}\underset{0}{\overset{1}{\xab}}{\mathrm{x}}^{2}\mathrm{dx}\right]\\ =\mathrm{}2\mathrm{x}\mathrm{}\frac{{\mathrm{x}}^{2}}{2}{\frac{{\mathrm{x}}^{3}}{3}}_{0}^{1}\\ =\mathrm{}2\mathrm{x}\left[\frac{1}{2}\frac{1}{3}\right]\\ =\mathrm{}2\mathrm{x}\left[\frac{1}{6}\right]\\ =\mathrm{}\frac{1}{3}\mathrm{sq}.\mathrm{units}\end{array}$
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CBSE Class 12 Maths Important Questions
FAQs (Frequently Asked Questions)
1. How does Important Questions of CBSE Class 12 Mathematics Chapter 8 help students in preparing for the board exam?
CBSE Class 12 Mathematics Chapter 8 Important Questions have been selected by subject matter experts after an extensive analysis of the questions repeated frequently in exams. Many students skip this chapter because of a lack of understanding of the concepts. Even after understanding the concepts clearly, students must extensively practise to solve the questions in a short time. This chapter is also important for entrance exams like the IIT JEE. Students should practise these questions to get a thorough understanding of the chapter and the questions that are important from the exam perspective.
2. How to practise important questions from CBSE Class 12 Mathematics Chapter 8?
Practice is important, especially for a subject like Mathematics. Therefore, students must practise more questions before the exam so that they can solve them in the minimum amount of time. A few points have been mentioned below that students should keep in mind while solving the questions.
 Read the question well to understand what exactly it is asking for.
 Try to solve the questions on your own. It is advised to look at the answers given only when you cannot solve the question or understand the steps needed.
 Try to finish every question within the time limit.
 After completing the question, evaluate it to determine which areas should be improved further.
 If you cannot solve a question on the first attempt, understand how it’s done and then try it several times until you become confident with the concept.
3. Is the chapter “Application of Integrals” important?
Chapter 8 “Application of Integrals” is a very important chapter for the final year students. This chapter is crucial for not only the CBSE Board exam but also other competitive exams as well. The analysis of the question papers of the past years shows that roughly six to eight marks’ questions are given from this chapter. Furthermore, this chapter is comparatively easy, and students can solve questions if concepts are clear to them. So, students should not take the chapter lightly; they should focus on practising questions of different types that are important from an exam perspective.
4. What are the sources for practising CBSE Class 12 Mathematics Chapter 8?
The biggest problem that students face nowadays is the abundance of study materials and question banks. There are several books and numerous websites providing every possible type of answer to the questions in this chapter. However, only a few of them have the right approach to excel in the board exams. Therefore, Class 12 NCERT Books should be the primary source for the students. Once they are wellversed with the concepts, they should look for past years’ questions and practise some model papers. Important Questions provided by Extramarks, is a collection of carefully chosen questions that maintains the standard of board exams. Students can refer to them if they want a set of questions that are relevant and handy before the exams.