RD Sharma Class 12 Exercise 20.2 Areas Of Bounded Region Solutions Maths - Download PDF Free Online

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# RD Sharma Class 12 Exercise 20.2 Areas Of Bounded Region Solutions Maths - Download PDF Free Online

Edited By Kuldeep Maurya | Updated on Jan 24, 2022 03:19 PM IST

Maths is a tricky subject to master because of the abundance of formulae and theorems. To get a good hold on the subject, students need to refer to explicit materials covering all concepts. This is where RD Sharma books come to help with their explicit material and easy-to-understand images.

RD Sharma Class 12th Exercise 20.2 is a relatively short exercise covering concepts like finding areas of curves through integration. It contains four Level 1 questions and 1 Level 2 question. RD Sharma solutions This exercise is based on the area of the region bounded by parabola and lines, parabola and its latus rectum. However, students can refer to the material provided by Career360 to complete this exercise in a breeze.

## Areas of Bounded Regions Exercise:20.2

Area of bounded region exercise 20.2 question 1

$\frac{7}{3}$ sq.units
Hint:
$y=4x^{2}$ is an upward parabola with vertex (0,0)
Given:
$y=4x^{2}$
Solution:

x=0 is the line with points (0,1),(0,0) etc.,
y=1 is the line with points (0,1),(1,1) etc.,
y=4 is the line with points (0,4),(1,4) etc.,
Parabola meets at points (0,0) with line x=0
$\left ( \frac{1}{2},1 \right )$ with liney=1, (1,4) with line y=4
Here we need to do along y-axis as we don’t know equation of line with respect to x
Required area (Shaded one)=Area under parabola from y=1 to y=4
\begin{aligned} &\int_{1}^{4}|x| d y=\int_{1}^{4} \sqrt{\frac{y}{4}} d y \\ \end{aligned}
\begin{aligned} &=\frac{1}{2} \int_{1}^{4} \sqrt{y} d y \\ &=\left[\frac{1}{2} \frac{y^{\frac{3}{2}}}{\frac{3}{2}}\right]_{1}^{4} \\ \end{aligned}
\begin{aligned} &=\frac{1}{2} \times \frac{2}{3}\left[y^{\frac{3}{2}}\right]_{1}^{4} \\ &=\frac{1}{3}\left[4^{\frac{3}{2}}-1^{\frac{3}{2}}\right] \\ \end{aligned}
\begin{aligned} &=\frac{1}{3}(8-1) \\ &=\frac{7}{3} s q \cdot u n \text { its } \end{aligned}

Area of bounded region exercise 20.2 question 2

$\frac{56}{3}sq.units$
Hint:
To find the area under two or more than two curves, the first crucial step is to find the intersection points of the curves.
Given:
$x^{2}=16y$, $y=1$, $y=4$and y-axis in the first quadrant
Solution:

$y=\frac{x^{2}}{16}$ (Curve)
$y=4$ (Line )
$y=1$ (Line)
Between curve A and line C
$16=x^{2}\\ x=4$
Between curve A and line B
$4=\frac{x^{2}}{16}\\ x=\sqrt{64}\\ x=8$
Required area can be calculated by breaking the problem into two parts
1. Calculate area under the curve A and Line B
2. Subtract the area enclosed by curve A and line C from the above area.
1. $\int_{0}^{8}\left(4-\frac{x^{2}}{16}\right) d x=$ Area under B and A
\begin{aligned} &=[4 x]_{0}^{8}-\left[\frac{x^{3}}{16(3)}\right]_{0}^{8} \\ &=\frac{64}{3} s q \cdot \text { units } \end{aligned}
2. $\int_{0}^{4}\left(1-\frac{x^{2}}{16}\right) d x=$ Area under C and A
\begin{aligned} &=[x]_{0}^{4}-\left[\frac{x^{3}}{16(3)}\right]_{0}^{4} \\ &=\frac{8}{3} s q \cdot \text { units } \end{aligned}
The required area under the curve
$\frac{64}{3}-\frac{8}{3} =\frac{56}{3}$ sq.units

Area of bounded region exercise 20.2 question 3

$\frac{1}{24a}$
Hint:
Find the latus rectum and its intersection points with the parabola and then integrate the expression to find the area enclosed by the curves.
Given:
$x^{2}=4ay$
Solution:

First, we find the latus rectum and its intersection points with the parabola
$x^{2}=4ay$ … (i)
$y=\frac{x^{2}}{4a}$ … (ii)
Comparing it with the standard from of a parabola
$Y^{2}=4aX^{2}$
Where (0,A) is the coordinate of the focus of the parabola and the latus rectum passes through this point and is perpendicular to the axis of symmetry
Therefore, the equation of the latus rectum is y=a
Comparing equation (i) and (ii)
$\frac{1}{4a}=4A$
$A=\frac{1}{16a}$
Intersection point
\begin{aligned} &\frac{1}{16 a}=\frac{x^{2}}{4 a} \\ &x^{2}=\frac{1}{4} \\ &x=\pm \frac{1}{2} \end{aligned}
Now, Integrating the expression to find the area enclosed by the curves
Since, the latus rectum is above the parabola in the Cartesian plane, the expression will be
\begin{aligned} &\int_{-\frac{1}{2}}^{\frac{1}{2}}\left[\frac{1}{16 a}-\frac{x^{2}}{4 a}\right] d x \\ &=\left[\frac{x}{16 a}\right]_{-\frac{1}{2}}^{\frac{1}{2}}-\left[\frac{x^{3}}{4 a(3)}\right]_{-\frac{1}{2}}^{\frac{1}{2}} \\ \end{aligned}
\begin{aligned} &=\frac{1}{16 a}-\frac{1}{48 a} \\ &=\frac{1}{24 a} \end{aligned}

Area of bounded region exercise 20.2 question 4

$\frac{1}{96}$Hint:
Find the latus rectum and its intersection points with the parabola and then integrate the expression to find the area enclosed by the curves.
Given:
$x^{2}+16y=0$
Solution:

Curve is $x^{2}+16y=0$
To find the enclosed area
\begin{aligned} &2 \times \int_{0}^{8}\left(\frac{-x^{2}}{16}-(-4)\right) d x \\ \end{aligned}
\begin{aligned} &=2 \times\left[\frac{-x^{3}}{16(3)}+4 x\right]_{0}^{8} \\ &=2 \times\left[\frac{8^{3}-0^{3}}{48}-4(8-0)\right] \\ \end{aligned}
\begin{aligned} &=2 \times\left[\frac{512}{48}-32\right] \\ &=2 \times\left[\frac{32}{3}-32\right] \\ \end{aligned}
\begin{aligned} &=2 \times \frac{64}{3} \\ &=\frac{128}{3} \end{aligned}
So, the answer is \begin{aligned} &\frac{128}{3} \end{aligned} sq. Units

Area of bounded region exercise 20.2 question 5

$\frac{3}{5}a^{2}\left ( 2^{\frac{5}{3}} -1\right )$
Hint:
Area of the bounded region = Area under (y=2a) and $(ay^{2}=x^{3})$- Area under $(y=2a)$ and $(ay^{2}=x^{2})$
Given: $ay^{2}=x^{3}, y=a, y=2a$
Solution:

Area under $(y=2a)$ and $(ay^{2}=x^{2})$

\begin{aligned} &=\int_{0}^{1.58 a}\left(2 a-\sqrt{\frac{x^{3}}{a}}\right) d x \\ \end{aligned}
\begin{aligned} &=\left[2 a x-\frac{2 x^{\frac{5}{2}}}{5 a^{\frac{1}{2}}}\right]_{0}^{1.58 a} \\ \end{aligned}
\begin{aligned} &=2 a(1.58 a)-\frac{2(1.58 a)^{\frac{5}{2}}}{5 a^{\frac{1}{2}}} \end{aligned}
Area under $(y=a)$ and $(ay^{2}=x^{2})$
\begin{aligned} &=\int_{0}^{a}\left(a-\sqrt{\frac{x^{2}}{a}}\right) d x \\ \end{aligned}
\begin{aligned} &=\left[a x-\frac{2 x^{\frac{5}{2}}}{5 a^{\frac{1}{2}}}\right]_{0}^{a} \\ \end{aligned}
\begin{aligned} &=a(a)-\frac{2(a)^{\frac{5}{2}}}{5 a^{\frac{1}{2}}} \end{aligned}
Required area
$2 a(1.58 a)-\frac{2(1.58 a)^{\frac{5}{2}}}{5 a^{\frac{1}{2}}}-\left[a^{2}-\frac{2(a)^{\frac{5}{2}}}{5 a^{\frac{1}{2}}}\right]$
Solving further,

\begin{aligned} &\frac{3}{5} a^{2}\left(2(2)^{\frac{2}{3}}-1\right) \text { sq. Units } \\ \end{aligned}
\begin{aligned} &=\frac{3}{5} a^{2}\left(2^{1+\frac{2}{3}}-1\right) \text { sq. Units } \\ \end{aligned}\begin{aligned} &=\frac{3}{5} a^{2}\left(2^{\frac{5}{3}}-1\right) \text { sq. Units } \end{aligned}

RD Sharma Class 12th Exercise 20.2 contains the best solutions for the RD Sharma book that helps students score well in exams. Maths deals with a lot of chapters that have hundreds of sums. Preparation before exams is challenging for students as they have to cover every chapter and its concepts.

To ease up the stress for students before exams, Career360 has prepared RD Sharma Class 12th Exercise 20.2 solved material. It covers the entire syllabus and is designed by experts who have years of experience on CBSE question papers. This is why students can be confident that this material will help them score better marks in their exams while saving time.

As students are given lots of homework in maths due to the abundance of questions, RD Sharma Class 12th Exercise 20.2 material will help them cover up their portion quickly and finish their assignment. The bonus of this material is that it complies with the CBSE syllabus. This means that students can compare their progress and stay on par with the class lectures.

Career360 has provided RD Sharma Class 12th Exercise 20.2 solutions as a guide for students during preparation. They can refer to it during difficulties in understanding questions from the textbook. Every solution is given with a step-by-step solution which means that students with a low academic performance also can pick up concepts quickly and learn effectively.

RD Sharma Class 12 Chapter 20 Exercise 20.2 material goes through a series of quality checks to ensure that students get the best quality answers. Students feel overwhelmed by the number of questions in the RD Sharma textbook and miss out on important topics. The best trick of learning Class 12 Maths is to divide up the work and then study regularly. This way, they won’t feel the pressure and complete the syllabus on time. Additionally, these solutions will help speed up the process even more as they won’t have to solve every question.

## RD Sharma Chapter wise Solutions

1. Who can use this material?

Class 12 RD Sharma Chapter 20 Exercise 20.2 Solutions material is made for Class 12 CBSE students to prepare for their exams. This is beneficial for students as it covers the entire syllabus.

2. Does this material help in finishing up homework?

As RD Sharma is a widely used book, teachers often use it for lectures and homework questions. Thus, RD Sharma Class 12 Solutions Areas of Bounded Region Ex 20.2 material can help students in solving their homework problems.

3. Are there any extra charges for this material?

There are no hidden charges for this material, and students can access it from Career360’s website for free.

4. What are the benefits of this material?

RD Sharma Class 12 Solutions Chapter 20 Ex 20.2 material is exam-oriented and contains step-by-step solutions for each question. Moreover, as it is prepared by experts and updated to the latest version, students can benefit by writing better answers in exams.

5. What is integration using limits?

Limits are the values that are substituted after integrating the function. For example, there are upper and lower limits for an integral, which must be subtracted in the end.

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##### Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
##### Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. Ever since internet cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

3 Jobs Available
##### Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available
##### Content Writer

Content writing is meant to speak directly with a particular audience, such as customers, potential customers, investors, employees, or other stakeholders. The main aim of professional content writers is to speak to their targeted audience and if it is not then it is not doing its job. There are numerous kinds of the content present on the website and each is different based on the service or the product it is used for.

2 Jobs Available
##### Reporter

Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.

2 Jobs Available
##### Linguist

Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).

Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian

2 Jobs Available
##### Production Manager

Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
##### Production Engineer

A career as Production Engineer is crucial in the manufacturing industry. He or she ensures the functionality of production equipment and machinery to improve productivity and minimize production costs in order to drive revenues and increase profitability.

2 Jobs Available
##### Automation Test Engineer

An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.

2 Jobs Available
##### Product Designer

Individuals who opt for a career as product designers are responsible for designing the components and overall product concerning its shape, size, and material used in manufacturing. They are responsible for the aesthetic appearance of the product. A product designer uses his or her creative skills to give a product its final outlook and ensures the functionality of the design.

Students can opt for various product design degrees such as B.Des and M.Des to become product designers. Industrial product designer prepares 3D models of designs for approval and discusses them with clients and other colleagues. Individuals who opt for a career as a product designer estimate the total cost involved in designing.

2 Jobs Available
##### R&D Personnel

A career as R&D Personnel requires researching, planning, and implementing new programs and protocols into their organization and overseeing new products’ development. He or she uses his or her creative abilities to improve the existing products as per the requirements of the target market.

2 Jobs Available
##### Commercial Manager

A Commercial Manager negotiates, advises and secures information about pricing for commercial contracts. He or she is responsible for developing financial plans in order to maximise the business's profitability.

2 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### ITSM Manager

ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

3 Jobs Available
##### Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available
##### Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
##### Computer System Analyst

Individuals in the computer systems analyst career path study the hardware and applications that are part of an organization's computer systems, as well as how they are used. They collaborate closely with managers and end-users to identify system specifications and business priorities, as well as to assess the efficiency of computer systems and create techniques to boost IT efficiency. Individuals who opt for a career as a computer system analyst support the implementation, modification, and debugging of new systems after they have been installed.

2 Jobs Available
##### Test Manager

A Test Manager is a professional responsible for planning, coordinating and controlling test activities. He or she develops test processes and strategies to analyse and determine test methods and tools for test activities. The test manager jobs involve documenting tests that have been carried out, analysing and evaluating software quality to determine further recommended procedures.

2 Jobs Available
##### Azure Developer

A career as Azure Developer comes with the responsibility of designing and developing cloud-based applications and maintaining software components. He or she possesses an in-depth knowledge of cloud computing and Azure app service.

2 Jobs Available
##### Deep Learning Engineer

A Deep Learning Engineer is an IT professional who is responsible for developing and managing data pipelines. He or she is knowledgeable about analyzing and storing data collected from various sources.  A Career as a Deep Learning Engineer needs to help the  data scientists and analysts to create effective data sets.

2 Jobs Available