This document contains sample problems and solutions for calculating the area of regions bounded by curves like lines, parabolas, circles, and ellipses using integral calculus. It includes 12 problems finding areas bounded by mathematical curves and determining values that divide regions into equal parts. The problems apply techniques like finding points of intersection and using integrals to calculate areas.
Important questions for class 10 maths chapter 3 pair of linear equations in ...ExpertClass
expert's class ahmedabad by dhruv thakar
important notes of mathematics class 10
ncert solution also available ,mathematics adda
top institute in ahmedabad for mathematics
school and And graduation level competitive exam mathematics
dhruv sir mobile number-8320495706 ,bhai ka mathematis
Dear Students/Parents
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Important questions for class 10 maths chapter 3 pair of linear equations in ...ExpertClass
expert's class ahmedabad by dhruv thakar
important notes of mathematics class 10
ncert solution also available ,mathematics adda
top institute in ahmedabad for mathematics
school and And graduation level competitive exam mathematics
dhruv sir mobile number-8320495706 ,bhai ka mathematis
Dear Students/Parents
APEX INSTITUTE has been established with sincere and positive resolve to do something rewarding for ENGG. / PRE-MEDICAL aspirants. For this the APEX INSTITUTE has been instituted to provide a relentlessly motivating and competitive atmosphere.
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
APEX INSTITUTE has been established with sincere and positive resolve to do something rewarding for ENGG. / PRE-MEDICAL aspirants. For this the APEX INSTITUTE has been instituted to provide a relentlessly motivating and competitive atmosphere.
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Grooming at the APEX INSTITUTE is done methodically focusing on understanding of the subject, tricks of tackling the questions and above all enthusing students with self confidence, ambition and a 'never say give up' spirit. As secrets of success these are no substitutes for hard work and patience. These qualities, the APEX INSTITUTE ensures will be developed in students in full measure perhaps luck plays some role in one's life and career. But it is also a universal truth that before proper grasps of subject. Intelligent preparation and perservance even adverse luck does not stand a chance.
With this mantra success is sure to come your way. At APEX INSTITUTE we strive our best to realize the Alchemist's dream of turning 'base metal' into 'gold'
Teaching Mathematics Concepts via Computer Algebra Systemsinventionjournals
Most articles examine computer algebra systems (CAS) as they relate to the teaching and
learning of mathematics from advantages to disadvantages. This paper will explore junior undergraduate
students’ ability to solve distinguish tricky examples using various CAS technologies. Additionally, an
understanding for how CAS technologies are adopted and applied in professional environments is valuable,
both in guiding improvements to these tools and identifying new tools which can aid mathematician
Dear Students/Parents
APEX INSTITUTE has been established with sincere and positive resolve to do something rewarding for ENGG. / PRE-MEDICAL aspirants. For this the APEX INSTITUTE has been instituted to provide a relentlessly motivating and competitive atmosphere.
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Grooming at the APEX INSTITUTE is done methodically focusing on understanding of the subject, tricks of tackling the questions and above all enthusing students with self confidence, ambition and a 'never say give up' spirit. As secrets of success these are no substitutes for hard work and patience. These qualities, the APEX INSTITUTE ensures will be developed in students in full measure perhaps luck plays some role in one's life and career. But it is also a universal truth that before proper grasps of subject. Intelligent preparation and perservance even adverse luck does not stand a chance.
With this mantra success is sure to come your way. At APEX INSTITUTE we strive our best to realize the Alchemist's dream of turning 'base metal' into 'gold'
Teaching Mathematics Concepts via Computer Algebra Systemsinventionjournals
Most articles examine computer algebra systems (CAS) as they relate to the teaching and
learning of mathematics from advantages to disadvantages. This paper will explore junior undergraduate
students’ ability to solve distinguish tricky examples using various CAS technologies. Additionally, an
understanding for how CAS technologies are adopted and applied in professional environments is valuable,
both in guiding improvements to these tools and identifying new tools which can aid mathematician
Transportation problem is one of the sub classes of Linear Programming Problem (LPP) in
which the objective is to transport various quantities of a single product that are stored at various
origins to several destinations in such a way that the total transportation cost is minimum. The costs
of shipping from sources to destinations are indicated by the entries in the matrix. To achieve this
objective we must know the amount and location of available supplies and the quantities demanded.
The different solution procedure of such type of problem illustrate in given paper.
WHY CHILDREN ABSORBS MORE MICROWAVE RADIATION THAT ADULTS: THE CONSEQUENCES” ...Shahrukh Javed
In today’s world, technologic developments bring social and economic benefits to large sections of society; however, the health consequences of these developments can be difficult to predict and manage. With rapid advancement in technologies and communications, children are increasingly exposed to MWRs at earlier and earlier ages. Microwave radiation causes damaging health effects, especially for children, whose brains are fragile and still developing. MWR penetration is greater relative to developing brain of fetus and child, and they will have a longer penetration of exposure than adults. This concerns the effects of microwave on health because they pervade diverse fields of our lives. Thus, children are at greater risk than adults when exposed to possible human carcinogen[1], computer simulation using MRI scans of children is the only possible way to determine the microwave radiation (MWR) absorbed in specific tissues in children and this radiation is measured in terms of specific absorption rate[2] (SAR). It also includes an assessment of the potential susceptibility of children to MWR and concludes with a recommendation for additional research in present legal limits for exposure to MWR and the development of precautionary policies in the face of scientific uncertainty.
• UNIVERSAL ASYNCHRONOUS RECEIVER TRANSMITTER
• UART is a device that has the capability to both receive and transmit serial data.
• A universal asynchronous receiver-transmitter is a computer hardware device for asynchronous serial communication in which the data format and transmission speeds are configurable.
A heart rate monitor is a personal monitoring device that allows one to measure one's heart rate in real time or record the heart rate for later study. It is largely used by performers of various types of physical exercise.
A universal asynchronous receiver-transmitter is a computer hardware device for asynchronous serial communication in which the data format and transmission speeds are configurable.
As the ages of mankind progressed there was a rapid change in the means of transportation with technology starting from the days of bullockcarts to the present days of planes,drones and space shuttles. Travel speed increased from miles/hr to the speed of light.
The source to destination became so closer to each others. There was an advancement in the field of the air traveling system with the help of airplane. But as the speed increases , the horror of air crash also introduced. Because at a height of 2000m and above if a plane crashes ,it will be a terror for any body. So to take the feed back of the various activities happens in the plane and record them engineers need a mechanism to record such activities .
With any airplane crash, there are many unanswered questions as to what brought the plane down. Investigators turn to the airplane's flight data recorder (FDR) and cockpit voice recorder (CVR), also known as "BLACK BOXES" for answers. In recent times these black boxes helped to know answers for many of issues regarding the crashes and may come to help to solve many cases in the future also.so these black boxes play a vital role in this modern world with the advancement of technology
What is wireless power transmission(WPT)?
Why is WPT?
History of WPT
Types of WPT
Techniques to transfer energy wirelessly
Advantages and disadvantages
Applications
Conclusion
References
The main objective of this current proposal is to make the recharging of the mobile phones independent of their manufacturer and battery make. In this paper a new proposal has been made so as to make the recharging of the mobile phones is done automatically as you talk in your mobile phone!
The microwave signal is transmitted from the transmitter along with the message signal using special kind of antennas called slotted wave guide antenna at a frequency is 2.45 GHz
Why children absorb more microwave radiation than adults the consequencesShahrukh Javed
In today’s world, technologic developments bring social and economic benefits to large sections of society; however, the health consequences of these developments can be difficult to predict and manage.
With rapid advancement in technologies and communications, children are increasingly exposed to MWRs at earlier ages which causes damaging health effects, especially whose brains are fragile and still developing.
Radiation exposure of cell phones & its impact on humansShahrukh Javed
Radiation Exposure Of Cell Phones & Its Impact On Humans.
Electromagnetic Radiation.
Radio Waves Emitted By The Cell Phone.
Cell Phone Base Station.
Antennas On Cell Tower Transmit In The Frequency Range.
Types Of Antennas –Transmission.
Types Of Base Stations-operators.
Possible Health Effects Of ELF-EMF/RFR Exposure Effects & Base Station.
Illness That Have Potential Links To Phone Radiation.
Awareness Among People.
Wealth Vs Health.
The Precautionary Principle.
Conclusion.
Radiation exposure of cell phones & its impact on humans.Shahrukh Javed
RADIATION EXPOSURE OF CELL PHONES & ITS IMPACT ON HUMANS
ELECTROMAGNETIC RADIATION
RADIO WAVES EMITTED BY THE CELL PHONE
CELL PHONE BASE STATION
ANTENNAS ON CELL TOWER TRANSMIT IN THE FREQUENCY RANGE
TYPES OF ANTENNAS –TRANSMISSION.
TYPES OF BASE STATIONS-OPERATORS.
POSSIBLE HEALTH EFFECTS OF ELF-EMF/RFR EXPOSURE EFFECTS & BASE STATION
ILLNESS THAT HAVE POTENTIAL LINKS TO PHONE RADIATION
AWARENESS AMONG PEOPLE
WEALTH VS HEALTH
THE PRECAUTIONARY PRINCIPLE
CONCLUSION
Black Box Corporation, also doing business as Black Box Network Services, is headquartered in the Pittsburgh suburb of Lawrence, Pennsylvania, United States. The company is a provider of communications products.
Automatic image mosaicing an approach based on fftShahrukh Javed
An Image mosaic is a composition generated from a sequence of images which can be obtained by understanding geometric relationships between images. The geometric relations are coordinate transformations that relate the different image coordinate systems. Image mosaic is a technique for creating images which cannot be created by a single frame of the camera, for example satellite imagery. The user gives a series of pictures with overlapping regions and the software returns one large image with all the pictures merged as accurately as possible. When it is impossible to capture the large image in one shot with available equipment, mosaicing process can be used to create an image montage from separately scanned pieces.
This paper goal is to create a script that will stitch two images together to create one larger image. These stitched images, become panoramic views which increase the visual aesthetics of a scene, and are widely sought out for posters, postcards, and other printed materials. This stitching will be performed using point correspondences between the two images.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
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Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
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Chapter 8 application_of_integrals
1. Class XII Chapter 8 – Application of Integrals Maths
Page 1 of 53
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Exercise 8.1
Question 1:
Find the area of the region bounded by the curve y2
= x and the lines x = 1, x = 4 and
the x-axis.
Answer
The area of the region bounded by the curve, y2
= x, the lines, x = 1 and x = 4, and the
x-axis is the area ABCD.
2. Class XII Chapter 8 – Application of Integrals Maths
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Question 2:
Find the area of the region bounded by y2
= 9x, x = 2, x = 4 and the x-axis in the first
quadrant.
Answer
The area of the region bounded by the curve, y2
= 9x, x = 2, and x = 4, and the x-axis
is the area ABCD.
3. Class XII Chapter 8 – Application of Integrals Maths
Page 3 of 53
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Question 3:
Find the area of the region bounded by x2
= 4y, y = 2, y = 4 and the y-axis in the first
quadrant.
Answer
The area of the region bounded by the curve, x2
= 4y, y = 2, and y = 4, and the y-axis
is the area ABCD.
4. Class XII Chapter 8 – Application of Integrals Maths
Page 4 of 53
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Question 4:
Find the area of the region bounded by the ellipse
Answer
The given equation of the ellipse, , can be represented as
It can be observed that the ellipse is symmetrical about x-axis and y-axis.
∴ Area bounded by ellipse = 4 × Area of OAB
5. Class XII Chapter 8 – Application of Integrals Maths
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Therefore, area bounded by the ellipse = 4 × 3π = 12π units
Question 5:
Find the area of the region bounded by the ellipse
Answer
The given equation of the ellipse can be represented as
It can be observed that the ellipse is symmetrical about x-axis and y-axis.
∴ Area bounded by ellipse = 4 × Area OAB
6. Class XII Chapter 8 – Application of Integrals Maths
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Therefore, area bounded by the ellipse =
Question 6:
Find the area of the region in the first quadrant enclosed by x-axis, line and the
circle
Answer
The area of the region bounded by the circle, , and the x-axis is the
area OAB.
7. Class XII Chapter 8 – Application of Integrals Maths
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The point of intersection of the line and the circle in the first quadrant is .
Area OAB = Area ∆OCA + Area ACB
Area of OAC
Area of ABC
Therefore, area enclosed by x-axis, the line , and the circle in the first
quadrant =
Question 7:
Find the area of the smaller part of the circle x2
+ y2
= a2
cut off by the line
Answer
8. Class XII Chapter 8 – Application of Integrals Maths
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The area of the smaller part of the circle, x2
+ y2
= a2
, cut off by the line, , is the
area ABCDA.
It can be observed that the area ABCD is symmetrical about x-axis.
∴ Area ABCD = 2 × Area ABC
9. Class XII Chapter 8 – Application of Integrals Maths
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Therefore, the area of smaller part of the circle, x2
+ y2
= a2
, cut off by the line, ,
is units.
Question 8:
The area between x = y2
and x = 4 is divided into two equal parts by the line x = a, find
the value of a.
Answer
The line, x = a, divides the area bounded by the parabola and x = 4 into two equal
parts.
10. Class XII Chapter 8 – Application of Integrals Maths
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∴ Area OAD = Area ABCD
It can be observed that the given area is symmetrical about x-axis.
⇒ Area OED = Area EFCD
11. Class XII Chapter 8 – Application of Integrals Maths
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From (1) and (2), we obtain
Therefore, the value of a is .
Question 9:
Find the area of the region bounded by the parabola y = x2
and
Answer
The area bounded by the parabola, x2
= y,and the line, , can be represented as
12. Class XII Chapter 8 – Application of Integrals Maths
Page 12 of 53
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The given area is symmetrical about y-axis.
∴ Area OACO = Area ODBO
The point of intersection of parabola, x2
= y, and line, y = x, is A (1, 1).
Area of OACO = Area ∆OAB – Area OBACO
⇒ Area of OACO = Area of ∆OAB – Area of OBACO
Therefore, required area = units
13. Class XII Chapter 8 – Application of Integrals Maths
Page 13 of 53
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(One Km from ‘Welcome’ Metro Station)
Question 10:
Find the area bounded by the curve x2
= 4y and the line x = 4y – 2
Answer
The area bounded by the curve, x2
= 4y, and line, x = 4y – 2, is represented by the
shaded area OBAO.
Let A and B be the points of intersection of the line and parabola.
Coordinates of point .
Coordinates of point B are (2, 1).
We draw AL and BM perpendicular to x-axis.
It can be observed that,
Area OBAO = Area OBCO + Area OACO … (1)
Then, Area OBCO = Area OMBC – Area OMBO
14. Class XII Chapter 8 – Application of Integrals Maths
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Similarly, Area OACO = Area OLAC – Area OLAO
Therefore, required area =
Question 11:
Find the area of the region bounded by the curve y2
= 4x and the line x = 3
Answer
The region bounded by the parabola, y2
= 4x, and the line, x = 3, is the area OACO.
15. Class XII Chapter 8 – Application of Integrals Maths
Page 15 of 53
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The area OACO is symmetrical about x-axis.
∴ Area of OACO = 2 (Area of OAB)
Therefore, the required area is units.
Question 12:
Area lying in the first quadrant and bounded by the circle x2
+ y2
= 4 and the lines x = 0
and x = 2 is
16. Class XII Chapter 8 – Application of Integrals Maths
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A. π
B.
C.
D.
Answer
The area bounded by the circle and the lines, x = 0 and x = 2, in the first quadrant is
represented as
Thus, the correct answer is A.
17. Class XII Chapter 8 – Application of Integrals Maths
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Question 13:
Area of the region bounded by the curve y2
= 4x, y-axis and the line y = 3 is
A. 2
B.
C.
D.
Answer
The area bounded by the curve, y2
= 4x, y-axis, and y = 3 is represented as
Thus, the correct answer is B.
18. Class XII Chapter 8 – Application of Integrals Maths
Page 18 of 53
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Exercise 8.2
Question 1:
Find the area of the circle 4x2
+ 4y2
= 9 which is interior to the parabola x2
= 4y
Answer
The required area is represented by the shaded area OBCDO.
Solving the given equation of circle, 4x2
+ 4y2
= 9, and parabola, x2
= 4y, we obtain the
point of intersection as .
It can be observed that the required area is symmetrical about y-axis.
∴ Area OBCDO = 2 × Area OBCO
We draw BM perpendicular to OA.
Therefore, the coordinates of M are .
Therefore, Area OBCO = Area OMBCO – Area OMBO
19. Class XII Chapter 8 – Application of Integrals Maths
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Therefore, the required area OBCDO is
units
Question 2:
Find the area bounded by curves (x – 1)2
+ y2
= 1 and x2
+ y 2
= 1
Answer
The area bounded by the curves, (x – 1)2
+ y2
= 1 and x2
+ y 2
= 1, is represented by
the shaded area as
20. Class XII Chapter 8 – Application of Integrals Maths
Page 20 of 53
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On solving the equations, (x – 1)2
+ y2
= 1 and x2
+ y 2
= 1, we obtain the point of
intersection as A and B .
It can be observed that the required area is symmetrical about x-axis.
∴ Area OBCAO = 2 × Area OCAO
We join AB, which intersects OC at M, such that AM is perpendicular to OC.
The coordinates of M are .
21. Class XII Chapter 8 – Application of Integrals Maths
Page 21 of 53
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Therefore, required area OBCAO = units
Question 3:
Find the area of the region bounded by the curves y = x2
+ 2, y = x, x = 0 and x = 3
Answer
The area bounded by the curves, y = x2
+ 2, y = x, x = 0, and x = 3, is represented by
the shaded area OCBAO as
22. Class XII Chapter 8 – Application of Integrals Maths
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Then, Area OCBAO = Area ODBAO – Area ODCO
Question 4:
Using integration finds the area of the region bounded by the triangle whose vertices are
(–1, 0), (1, 3) and (3, 2).
Answer
BL and CM are drawn perpendicular to x-axis.
It can be observed in the following figure that,
Area (∆ACB) = Area (ALBA) + Area (BLMCB) – Area (AMCA) … (1)
23. Class XII Chapter 8 – Application of Integrals Maths
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Equation of line segment AB is
Equation of line segment BC is
Equation of line segment AC is
Therefore, from equation (1), we obtain
24. Class XII Chapter 8 – Application of Integrals Maths
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Area (∆ABC) = (3 + 5 – 4) = 4 units
Question 5:
Using integration find the area of the triangular region whose sides have the equations y
= 2x +1, y = 3x + 1 and x = 4.
Answer
The equations of sides of the triangle are y = 2x +1, y = 3x + 1, and x = 4.
On solving these equations, we obtain the vertices of triangle as A(0, 1), B(4, 13), and C
(4, 9).
It can be observed that,
Area (∆ACB) = Area (OLBAO) –Area (OLCAO)
Question 6:
Smaller area enclosed by the circle x2
+ y2
= 4 and the line x + y = 2 is
A. 2 (π – 2)
25. Class XII Chapter 8 – Application of Integrals Maths
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B. π – 2
C. 2π – 1
D. 2 (π + 2)
Answer
The smaller area enclosed by the circle, x2
+ y2
= 4, and the line, x + y = 2, is
represented by the shaded area ACBA as
It can be observed that,
Area ACBA = Area OACBO – Area (∆OAB)
Thus, the correct answer is B.
Question 7:
Area lying between the curve y2
= 4x and y = 2x is
A.
26. Class XII Chapter 8 – Application of Integrals Maths
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B.
C.
D.
Answer
The area lying between the curve, y2
= 4x and y = 2x, is represented by the shaded
area OBAO as
The points of intersection of these curves are O (0, 0) and A (1, 2).
We draw AC perpendicular to x-axis such that the coordinates of C are (1, 0).
∴ Area OBAO = Area (∆OCA) – Area (OCABO)
27. Class XII Chapter 8 – Application of Integrals Maths
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Thus, the correct answer is B.
28. Class XII Chapter 8 – Application of Integrals Maths
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Miscellaneous Solutions
Question 1:
Find the area under the given curves and given lines:
(i) y = x2
, x = 1, x = 2 and x-axis
(ii) y = x4
, x = 1, x = 5 and x –axis
Answer
i. The required area is represented by the shaded area ADCBA as
ii. The required area is represented by the shaded area ADCBA as
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Question 2:
Find the area between the curves y = x and y = x2
Answer
The required area is represented by the shaded area OBAO as
30. Class XII Chapter 8 – Application of Integrals Maths
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The points of intersection of the curves, y = x and y = x2
, is A (1, 1).
We draw AC perpendicular to x-axis.
∴ Area (OBAO) = Area (∆OCA) – Area (OCABO) … (1)
Question 3:
Find the area of the region lying in the first quadrant and bounded by y = 4x2
, x = 0, y
= 1 and y = 4
31. Class XII Chapter 8 – Application of Integrals Maths
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Answer
The area in the first quadrant bounded by y = 4x2
, x = 0, y = 1, and y = 4 is
represented by the shaded area ABCDA as
Question 4:
Sketch the graph of and evaluate
Answer
32. Class XII Chapter 8 – Application of Integrals Maths
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The given equation is
The corresponding values of x and y are given in the following table.
x – 6 – 5 – 4 – 3 – 2 – 1 0
y 3 2 1 0 1 2 3
On plotting these points, we obtain the graph of as follows.
It is known that,
33. Class XII Chapter 8 – Application of Integrals Maths
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Question 5:
Find the area bounded by the curve y = sin x between x = 0 and x = 2π
Answer
The graph of y = sin x can be drawn as
∴ Required area = Area OABO + Area BCDB
Question 6:
Find the area enclosed between the parabola y2
= 4ax and the line y = mx
Answer
The area enclosed between the parabola, y2
= 4ax, and the line, y = mx, is represented
by the shaded area OABO as
34. Class XII Chapter 8 – Application of Integrals Maths
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The points of intersection of both the curves are (0, 0) and .
We draw AC perpendicular to x-axis.
∴ Area OABO = Area OCABO – Area (∆OCA)
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Question 7:
Find the area enclosed by the parabola 4y = 3x2
and the line 2y = 3x + 12
Answer
The area enclosed between the parabola, 4y = 3x2
, and the line, 2y = 3x + 12, is
represented by the shaded area OBAO as
The points of intersection of the given curves are A (–2, 3) and (4, 12).
We draw AC and BD perpendicular to x-axis.
∴ Area OBAO = Area CDBA – (Area ODBO + Area OACO)
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Question 8:
Find the area of the smaller region bounded by the ellipse and the line
Answer
The area of the smaller region bounded by the ellipse, , and the line,
, is represented by the shaded region BCAB as
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∴ Area BCAB = Area (OBCAO) – Area (OBAO)
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Question 9:
Find the area of the smaller region bounded by the ellipse and the line
Answer
The area of the smaller region bounded by the ellipse, , and the line,
, is represented by the shaded region BCAB as
∴ Area BCAB = Area (OBCAO) – Area (OBAO)
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Question 10:
Find the area of the region enclosed by the parabola x2
= y, the line y = x + 2 and x-
axis
Answer
The area of the region enclosed by the parabola, x2
= y, the line, y = x + 2, and x-axis
is represented by the shaded region OABCO as
40. Class XII Chapter 8 – Application of Integrals Maths
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The point of intersection of the parabola, x2
= y, and the line, y = x + 2, is A (–1, 1).
∴ Area OABCO = Area (BCA) + Area COAC
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Question 11:
Using the method of integration find the area bounded by the curve
[Hint: the required region is bounded by lines x + y = 1, x – y = 1, – x + y = 1 and – x
– y = 11]
Answer
The area bounded by the curve, , is represented by the shaded region ADCB
as
The curve intersects the axes at points A (0, 1), B (1, 0), C (0, –1), and D (–1, 0).
It can be observed that the given curve is symmetrical about x-axis and y-axis.
∴ Area ADCB = 4 × Area OBAO
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Question 12:
Find the area bounded by curves
Answer
The area bounded by the curves, , is represented by the
shaded region as
It can be observed that the required area is symmetrical about y-axis.
Question 13:
Using the method of integration find the area of the triangle ABC, coordinates of whose
vertices are A (2, 0), B (4, 5) and C (6, 3)
43. Class XII Chapter 8 – Application of Integrals Maths
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Answer
The vertices of ∆ABC are A (2, 0), B (4, 5), and C (6, 3).
Equation of line segment AB is
Equation of line segment BC is
Equation of line segment CA is
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Area (∆ABC) = Area (ABLA) + Area (BLMCB) – Area (ACMA)
Question 14:
Using the method of integration find the area of the region bounded by lines:
2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0
Answer
The given equations of lines are
2x + y = 4 … (1)
3x – 2y = 6 … (2)
And, x – 3y + 5 = 0 … (3)
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The area of the region bounded by the lines is the area of ∆ABC. AL and CM are the
perpendiculars on x-axis.
Area (∆ABC) = Area (ALMCA) – Area (ALB) – Area (CMB)
Question 15:
Find the area of the region
Answer
The area bounded by the curves, , is represented as
46. Class XII Chapter 8 – Application of Integrals Maths
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The points of intersection of both the curves are .
The required area is given by OABCO.
It can be observed that area OABCO is symmetrical about x-axis.
∴ Area OABCO = 2 × Area OBC
Area OBCO = Area OMC + Area MBC
Question 16:
Area bounded by the curve y = x3
, the x-axis and the ordinates x = –2 and x = 1 is
A. – 9
B.
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C.
D.
Answer
Thus, the correct answer is B.
Question 17:
The area bounded by the curve , x-axis and the ordinates x = –1 and x = 1 is
given by
[Hint: y = x2
if x > 0 and y = –x2
if x < 0]
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A. 0
B.
C.
D.
Answer
Thus, the correct answer is C.
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Question 18:
The area of the circle x2
+ y2
= 16 exterior to the parabola y2
= 6x is
A.
B.
C.
D.
Answer
The given equations are
x2
+ y2
= 16 … (1)
y2
= 6x … (2)
Area bounded by the circle and parabola
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Area of circle = π (r)2
= π (4)2
= 16π units
Thus, the correct answer is C.
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Question 19:
The area bounded by the y-axis, y = cos x and y = sin x when
A.
B.
C.
D.
Answer
The given equations are
y = cos x … (1)
And, y = sin x … (2)
Required area = Area (ABLA) + area (OBLO)
Integrating by parts, we obtain
52. Class XII Chapter 8 – Application of Integrals Maths
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Thus, the correct answer is B.
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Therefore, the required area is units