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NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals are discussed here. These NCERT solutions are created by expert team at Creers360 keeping in mind of latest syllabus of CBSE 2023-24. In geometry, you must have learned some formulas to calculate areas of simple geometrical figures like triangles, rectangles, trapeziums, circles, etc. But how do you calculate areas enclosed by curves? In this article, you will get NCERT solutions for class 12 maths chapter 8 application of integrals. This article also includes application of integrals class 12 solutions. Important topics that are going to be discussed in this chapter 8 class 12 maths are the area under simple curves, the area between lines and arcs of circles, parabolas, and ellipses. Interested students can find all NCERT Solutions for Class 12 Maths in one place
In the applications of integrals class 12 ncert solutions, questions from all these topics are covered. In this composition of Class 12 Maths Chapter 8 NCERT solutions application of integrals, you will learn some important applications of integrals class 12. If you are interested to check all NCERT solutions from classes 6 to 12 in a single place, which will help you to learn CBSE maths and science. Here you will get NCERT solutions for class 12.
Also read:
>> Area Enclosed by a Curve and Lines: The area enclosed by the curve y = f(x), the x-axis, and the lines x = a and x = b (where b > a) is given by the formula:
Area = ∫[a, b]y.dx = ∫[a, b]f(x).dx
>> Area Bounded by Curve and Horizontal Lines: The area of the region bounded by the curve x = φ(y) as its y-axis and the lines y = c and y = d is given by the formula: Area = ∫[c, d]x.dy = ∫[c, d]φ(y).dy
>> Area Between Two Curves and Vertical Lines: The area enclosed between two given curves y = f(x) and y = g(x), and the lines x = a and x = b is given by the formula:
Area = ∫[a, b][f(x) - g(x)].dx (Where f(x) ≥ g(x) in [a, b])
>> Area Between Curves with Different Intervals: If f(x) ≥ g(x) in [a, c] and f(x) ≤ g(x) in [c, b], where a < c < b, then the resultant area between the curves is given as:
Area = ∫[a, c][f(x) - g(x)].dx + ∫[c, b][g(x) - f(x)].dx
Free download Application Of Integrals Class 12 NCERT Solutions for CBSE Exam.
NCERT class 12 maths chapter 8 question answer Exercise: 8.1
Question:1 Find the area of the region bounded by the curve and the lines and the -axis in the first quadrant.
Answer:
Area of the region bounded by the curve and the lines and the -axis in the first quadrant
Area =
= 14/3 units
Question:2 Find the area of the region bounded by and the -axis in the first quadrant.
Answer:
Area of the region bounded by the curve and the -axis in the first quadrant
Area =
units
Question:3 Find the area of the region bounded by and the -axis in the first quadrant.
Answer:
The area bounded by the curves and the -axis in the first quadrant is ABCD.
Question:4 Find the area of the region bounded by the ellipse
Answer:
The area bounded by the ellipse :
Area will be 4 times the area of EAB.
Therefore,
Therefore the area bounded by the ellipse will be
Question: 5 Find the area of the region bounded by the ellipse
Answer:
The area bounded by the ellipse :
The area will be 4 times the area of EAB.
Therefore,
Therefore the area bounded by the ellipse will be
Question: 6 Find the area of the region in the first quadrant enclosed by -axis, line and the circle
Answer:
The area of the region bounded by and is ABC shown:
The point B of the intersection of the line and the circle in the first quadrant is .
Area ABC = Area ABM + Area BMC where, M is point in x-axis perpendicular drawn from the line.
Now,area of ............(1)
and Area of
..................................(2)
then adding the area (1) and (2), we have then
The Area under ABC
Question: 7 Find the area of the smaller part of the circle cut off by the line
Answer:
we need to find the area of smaller part of the circle
Now,
Area of ABCD = 2 X Area of ABC
Area of ABC =
Area of ABCD = 2 X Area of ABC
Therefore, the area of the smaller part of the circle is
Question:8 The area between and is divided into two equal parts by the line , find the value of .
Answer:
we can clearly see that given area is symmetrical about x - axis
It is given that
Area of OED = Area of EFCD
Area of OED =
Area of EFCD =
Area of OED = Area of EFCD
Therefore, the value of a is
Question:9 Find the area of the region bounded by the parabola and .
Answer:
We can clearly see that given area is symmetrical about y-axis
Therefore,
Area of OCAO = Area of OBDO
Point of intersection of is (1 , 1) and (-1 , 1)
Now,
Area od OCAO = Area OAM - Area of OCMO
Area of OAM =
Area of OCMO =
Therefore,
Area od OCAO
Now,
Area of the region bounded by the parabola and is = 2 X Area od OCAO Units
Question: 10 Find the area bounded by the curve and the line .
Answer:
Points of intersections of is
Now,
Area of OBAO = Area of OBCO + Area of OCAO
Area of OBCO = Area of OMBCO- Area of OMBO
Area of OMBCO =
Area of OMBO =
Area of OBCO = Area of OMBCO- Area of OMBO
Similarly,
Area of OCAO = Area of OCALO - Area of OALO
Area of OCALO =
Area of OALO =
Area of OCAO = Area of OCALO - Area of OALO
Now,
Area of OBAO = Area of OBCO + Area of OCAO
Therefore, area bounded by the curve and the line is
Question: 11 Find the area of the region bounded by the curve and the line .
Answer:
The combined figure of the curve and
The required are is OABCO, and it is symmetrical about the horizontal axis.
Therefore, Area of OABCO = 2 Area of OAB
therefore the required area is units.
Question: 12 Choose the correct answer in the following
Area lying in the first quadrant and bounded by the circle and the lines and is
Answer:
The correct answer is A
The area bounded by circle C(0,0,4) and the line x=2 is
The required area = area of OAB
Question: 13 Choose the correct answer in the following.
Area of the region bounded by the curve , -axis and the line is
Answer:
The area bounded by the curve and y =3
the required area = OAB =
NCERT application of integrals class 12 solutions Exercise: 8.2
Question: 1 Find the area of the circle which is interior to the parabola .
Answer:
The area bounded by the circle and the parabola .
By solving the equation we get the intersecting point and
So, the required area (OBCDO)=2 times the area of (OBCO)
Draw a normal on the x-axis (M = )
Thus the area of OBCO = Area of OMBCO - Area of OMBO
S0, total area =
Question:2 Find the area bounded by curves and .
Answer:
Given curves are and
Point of intersection of these two curves are
and
We can clearly see that the required area is symmetrical about the x-axis
Therefore,
Area of OBCAO = 2 Area of OCAO
Now, join AB such that it intersects the x-axis at M and AM is perpendicular to OC
Coordinates of M =
Now,
Area OCAO = Area OMAO + Area CMAC
Now,
Area of OBCAO = 2 Area of OCAO
Therefore, the answer is
Question: 3 Find the area of the region bounded by the curves and .
Answer:
The area of the region bounded by the curves,
and is represented by the shaded area OCBAO as
Then, Area OCBAO will be = Area of ODBAO - Area of ODCO
which is equal to
Question: 4 Using integration find the area of region bounded by the triangle whose vertices are and .
Answer:
So, we draw BL and CM perpendicular to x-axis.
Then it can be observed in the following figure that,
We have the graph as follows:
Equation of the line segment AB is:
or
Therefore we have Area of
So, the equation of line segment BC is
or
Therefore the area of BLMCB will be,
Equation of the line segment AC is,
or
Therefore the area of AMCA will be,
Therefore, from equations (1), we get
The area of the triangle
Question:5 Using integration find the area of the triangular region whose sides have the equations and .
Answer:
The equations of sides of the triangle are .
ON solving these equations, we will get the vertices of the triangle as
Thus it can be seen that,
Question:6 Choose the correct answer.
Smaller area enclosed by the circle and the line is
Answer:
So, the smaller area enclosed by the circle, , and the line, , is represented by the shaded area ACBA as
Thus it can be observed that,
Area of ACBA = Area OACBO - Area of
Thus, the correct answer is B.
Question:7 Choose the correct answer.
Area lying between the curves and is
Answer:
The area lying between the curve, and is represented by the shaded area OBAO as
The points of intersection of these curves are and .
So, we draw AC perpendicular to x-axis such that the coordinates of C are (1,0).
Therefore the Area OBAO =
Thus the correct answer is B.
NCERT application of integrals class 12 solutions Miscellaneous: Exercise
Question:1 Find the area under the given curves and given lines:
Answer:
The area bounded by the curve and -axis
The area of the required region = area of ABCD
Hence the area of shaded region is 7/3 units
Question:1 Find the area under the given curves and given lines:
Answer:
The area bounded by the curev and -axis
The area of the required region = area of ABCD
Hence the area of the shaded region is 624.8 units
Question:2 Find the area between the curves and .
Answer:
the area between the curves and .
The curves intersect at A(1,1)
Draw a normal to AC to OC(x-axis)
therefore, the required area (OBAO)= area of (OCAO) - area of (OCABO)
Thus the area of shaded region is 1/6 units
Question:3 Find the area of the region lying in the first quadrant and bounded by and .
Answer:
the area of the region lying in the first quadrant and bounded by and .
The required area (ABCD) =
The area of the shaded region is 7/3 units
Question:4 Sketch the graph of and evaluate
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)
Integral to be evaluated is
Question:5 Find the area bounded by the curve between and .
Answer:
The graph of y=sinx is as follows
We need to find the area of the shaded region
ar(OAB)+ar(BCD)
=2ar(OAB)
The bounded area is 4 units.
Question:6 Find the area enclosed between the parabola and the line .
Answer:
We have to find the area of the shaded region OBA
The curves y=mx and y ^{ 2 } =4ax intersect at the following points
The required area is
Question:7 Find the area enclosed by the parabola and the line .
Answer:
We have to find the area of the shaded region COB
The two curves intersect at points (2,3) and (4,12)
Required area is
Question:8 Find the area of the smaller region bounded by the ellipse and the line .
Answer:
We have to find the area of the shaded region
The given ellipse and the given line intersect at following points
Since the shaded region lies above x axis we take y to be positive
The required area is
Question:9 Find the area of the smaller region bounded by the ellipse and the line .
Answer:
The area of the shaded region ACB is to be found
The given ellipse and the line intersect at following points
Y will always be positive since the shaded region lies above x axis
The required area is
Question:10 Find the area of the region enclosed by the parabola the line and the -axis.
Answer:
We have to find the area of the shaded region BAOB
O is(0,0)
The line and the parabola intersect in the second quadrant at (-1,1)
The line y=x+2 intersects the x axis at (-2,0)
The area of the region enclosed by the parabola the line and the -axis is 5/6 units.
Question:11 Using the method of integration find the area bounded by the curve
[ Hint: The required region is bounded by lines and ]
Answer:
We need to find the area of the shaded region ABCD
ar(ABCD)=4ar(AOB)
Coordinates of points A and B are (0,1) and (1,0)
Equation of line through A and B is y=1-x
The area bounded by the curve is 2 units.
Question:12 Find the area bounded by curves .
Answer:
We have to find the area of the shaded region
In the first quadrant
y=|x|=x
Area of the shaded region=2ar(OADO)
The area bounded by the curves is 1/3 units.
Question:13 Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are
Answer:
Equation of line joining A and B is
Equation of line joining B and C is
Equation of line joining A and C is
ar(ABC)=ar(ABL)+ar(LBCM)-ar(ACM)
ar(ABC)=8+5-6=7
Therefore the area of the triangle ABC is 7 units.
Question:14 Using the method of integration find the area of the region bounded by lines:
Answer:
We have to find the area of the shaded region ABC
ar(ABC)=ar(ACLM)-ar(ALB)-ar(BMC)
The lines intersect at points (1,2), (4,3) and (2,0)
Area of the region bounded by the lines is 3.5 units
Question:15 Find the area of the region .
Answer:
We have to find the area of the shaded region OCBAO
Ar(OCBAO)=2ar(OCBO)
For the fist quadrant
In the first quadrant, the curves intersect at a point
Area of the unshaded region in the first quadrant is
The total area of the shaded region is-
= Area of half circle - area of the shaded region in the first quadrant
Question:16 Choose the correct answer.
Area bounded by the curve , the -axis and the ordinates and is
Answer:
Hence the required area
Therefore the correct answer is B.
Question:17 Choose the correct answer.
T he area bounded by the curve , -axis and the ordinates and is given by
Answer:
The required area is
Question:18 Choose the correct answer.
The area of the circle exterior to the parabola is
Answer:
The area of the shaded region is to be found.
Required area =ar(DOC)+ar(DOA)
The region to the left of the y-axis is half of the circle with radius 4 units and centre origin.
Area of the shaded region to the left of y axis is ar(1) =
For the region to the right of y-axis and above x axis
The parabola and the circle in the first quadrant intersect at point
Remaining area is 2ar(2) is
Total area of shaded region is
Question:19 Choose the correct answer The area bounded by the -axis, and when is
Answer:
Given : and
Area of shaded region = area of BCDB + are of ADCA
Hence, the correct answer is B.
If you are looking for exercises solutions for chapter application of integrals class 12 then these are listed below.
Generally, one question (5 marks) is asked from this ch 8 maths class 12 in the 12th board final exam and you can score these 5 marks very easily with help of these NCERT solutions for class 12 maths chapter 8 application of integrals. In this chapter 8 class 12 maths, there are 2 exercises with 20 questions. In the NCERT class 12 maths ch 8 question answer, these questions are prepared and explained in a detailed manner using diagrams.
There are a total of 14 solved examples are given in the NCERT textbook to give a better understanding of the concepts related to the application of integrals class 12. Applications of integrals class 12 are indispensable for the Board exam. At the end of the chapter, 19 questions are given in a miscellaneous exercise. In the NCERT solutions for class 12 maths chapter 8 application of integrals article, you will get solutions to miscellaneous exercises too.
8.1 Introduction
8.2 Area under Simple Curves
8.2.1 The area of the region bounded by a curve and a line
8.3 Area between Two Curves
Also read,
Let's understand these topics with help of examples
The application of integrals class 12 ncert solutions offers several key features, including
Expertly crafted by knowledgeable subject specialists.
Designed to enhance students' fundamental understanding of the applications of integrals.
Clear and organized material.
All questions are answered using informative diagrams.
Supports students in completing their assignments and preparing for competitive exams.
Benefits of NCERT solutions:
Happy Reading !!!
Generally, one question of 5 marks is asked from this chapter in the 12th board final exam but if you want to obtain the full 5 five marks that demands practice, good strategy, and command of concepts, and therefore NCERT textbooks, NCERT syllabus becomes very important. it is advised for students to precisely go through the basics of class 12 maths chapter 8 question answer to score good marks.
Only knowing the answer is not enough to score good marks in the exam. One should know how to write the answer to board exam in order to get good marks. NCERT solutions are provided by experts who knows how best to write the answer in the board exam in order to get good marks. Practice class 12 maths chapter 8 question answer to command the concepts.
The NCERT Solutions for the application of integration class 12 at Careers360 are created by a team of experts through extensive research on each concept. The solutions are presented in a thorough and comprehensive manner, enabling students to perform well on class exams and board exams. Additionally, the solutions aid students in completing their assignments efficiently and on time. Interested students can download the application of integrals class 12 pdf.
Here you will get the detailed and comprehensive NCERT Solutions for class 12 maths by clicking on the link. if you want to study other material like NCERT Syllabus, NCERT Notes, NCERT Exercise Solutions, etc then you can browse the careers360 official website.
Some applications of integrals like finding area under simple curves, area of the region bounded by a curve and a line, and area between two curves are the important topics covered in this chapter. practice class 12 maths chapter 8 ncert solutions to score well in the exam.
The class 12 application of integrals ncert solutions can be challenging as it incorporates numerous formulas and intricate concepts. To gain a solid understanding of each topic, students must regularly test their mathematical skills by practicing the questions provided. With consistent revision and diligent effort, students can confidently ace any exam, be it for their board exams or in a competitive setting.
Application Date:09 September,2024 - 14 November,2024
Application Date:09 September,2024 - 14 November,2024
Admit Card Date:04 October,2024 - 29 November,2024
Admit Card Date:04 October,2024 - 29 November,2024
Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.
Possible steps:
Re-evaluate Your Study Strategies:
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Remember: This is a temporary setback. With the right approach and perseverance, you can overcome this challenge and achieve your goals.
I hope this information helps you.
Hi,
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hello mahima,
If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.
hope this helps.
Hello Akash,
If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.
You can get the Previous Year Questions (PYQs) on the official website of the respective board.
I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.
Thank you and wishing you all the best for your bright future.
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