RD Sharma Class 12 Exercise 20.4 Areas under the curve Solutions Maths - Download PDF Free Online

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# RD Sharma Class 12 Exercise 20.4 Areas under the curve Solutions Maths - Download PDF Free Online

Edited By Team Careers360 | Updated on Jan 29, 2022 11:41 AM IST

The Class 12 RD Sharma chapter 20 exercise 20.4 solution deals with the chapter of 'Area of bounded region,' which brings out the concepts of a bounded region which is a very essential concept for students if they have to understand the basic maths in a further road of knowledge. The RD Sharma class 12 exercise 20.4 collects the most critical questions important in this chapter.

## Areas of Bounded Regions Excercise: 20.4

Area of bounded region exercise 20.4 question 1

$\frac{32}{3}$ sq.units
Hint:
Find the point of intersection of the parabola and the given line and then find the required region.
Given:
$x=4y-y^{2}$and $x=2y-3$
Solution:
To find the point at the intersection of the parabola $x=4y-y^{2}$and the line $x=2y-3$

Let us substitute $x=2y-3$ in the equation of the parabola
\begin{aligned} &2 y-3=4 y-y^{2} \\ &y^{2}-2 y-3=0 \\ &(y+1)(y-3)=0 \\ &y=-1,3 \end{aligned}
Therefore, the point of intersection are D(-1,-5) and A(3,3)
The area of the required region ABCDOA,
$A=\int_{-1}^{3}\left | x_{1}-x_{2} \right |dy$where $x=4y-y^{2}$and $x=2y-3$
\begin{aligned} &=\int_{-1}^{3}\left(x_{1}-x_{2}\right) d y \quad\left[\because x_{1}>x_{2}\right] \\ &=\int_{-1}^{3}\left[\left(4 y-y^{2}\right)-(2 y-3)\right] d y \\ \end{aligned}
\begin{aligned} &=\int_{-1}^{3}\left[4 y-y^{2}-2 y+3\right] d y \\ &=\int_{-1}^{3}\left[-y^{2}+2 y+3\right] d y \\ \end{aligned}

\begin{aligned} &=\left[-3^{2}+3^{2}+9-\frac{1}{3}-1+3\right] \\ &=11-\frac{1}{3} \\ &=\frac{32}{3} s q \cdot \text { units } \end{aligned}

Area of bounded region exercise 20.4 question 3.1

9 sq. units
Hint:
Find the point of intersection of parabola and the given line and then integrate it to find the required region.
Given:
$y=2x-4$ and $y^{2}=4x$
Solution:

Substitute $y=2x-4$ in $y^{2}=4x$
\begin{aligned} &(2 x-4)^{2}=4 x \\ &4 x^{2}+16-16 x=4 x \\ &4 x^{2}-20 x+16=0 \\ \end{aligned}
\begin{aligned} &x^{2}-5 x+4=0 \\ &(x-1)(x-4)=0 \\ &x=1,4 \\ &y=-2,4 \end{aligned}
Therefore point of intersection are (1,-2) and (4,4)
Using horizontal strips:
The required area is the shaded region
\begin{aligned} A &=\int_{-2}^{4}\left(x_{1}-x_{2}\right) d y \end{aligned} where \begin{aligned} x_{1}=\frac{y+4}{2} \end{aligned} and \begin{aligned} x_{2}=\frac{y^{2}}{4} \\ \end{aligned}
\begin{aligned} &=\int_{-2}^{4}\left[\left(\frac{y+4}{2}\right)-\left(\frac{y^{2}}{4}\right)\right] d y \\ \end{aligned}

\begin{aligned} &=\left[\frac{4^{2}}{4}+2(4)-\frac{4^{3}}{12}\right]-\left[\frac{(-2)^{2}}{4}+2(-2)-\frac{(-2)^{3}}{12}\right] \\ \end{aligned}

Area of bounded region exercise 20.4 question 4

18 sq.units
Hint:
Find the point of intersection of parabola and the given line and then integrate it to find the required region.
Given:
x-y=4 and $y^{2}=2x$
Solution:

The parabola $y^{2}=2x$ opens towards the positive x-axis and its focus is $\left ( \frac{1}{2},0 \right )$
The straight line$x-y=4$ passes through (4,0) and (0,-4)
Solving $x-y=4$ and $y^{2}=2x$, we get
\begin{aligned} &y^{2}=2(y+4) \\ &y^{2}-2 y-8=0 \\ &(y-4)(y+2)=0 \\ &y=4,-2 \end{aligned}
So, the points of intersection of given parabola and the line are(8,4) and (2,-2)
Required area,
\begin{aligned} &=\int_{-2}^{4} x_{\text {line }} d y-\int_{-2}^{4} x_{\text {parabola }} d y \\ \end{aligned}
\begin{aligned} &=\int_{-2}^{4}(y+4) d y-\int_{-2}^{4} \frac{y^{2}}{2} d y \\ \end{aligned}
9\begin{aligned} &=\frac{(y+4)^{2}}{2}-\left[\frac{1}{2}\left(\frac{y^{3}}{3}\right)\right]_{-2}^{4} \\ \end{aligned}
\begin{aligned} &=\frac{1}{2}(64-6)-\frac{1}{6}[64-(-8)] \\ &=30-12 \\ &=18 \mathrm{sq} \cdot \text { units } \end{aligned}

## Area of bounded region exercise 20.4 question 3.2

$\frac{23}{3}$Hint:
Find the point of intersection of parabola and the given line and then integrate it to find the required region.
Given:
$y=2x-4$ and $y^{2}=4x$
Solution:

Substitute $y=2x-4$ in $y^{2}=4x$
\begin{aligned} &(2 x-4)^{2}=4 x \\ &4 x^{2}+16-16 x=4 x \\ &4 x^{2}-20 x+16=0 \\ \end{aligned}
\begin{aligned} &x^{2}-5 x+4=0 \\ &(x-1)(x-4)=0 \\ &x=1,4 \\ &y=-2,4 \end{aligned}
Therefore point of intersection are (1,-2) and (4,4)
Using vertical strips:
The required area is the shaded region
$A=\int_{0}^{4}y_{2}dx-\int_{1}^{4}y_{1}dx$where $y_{2}=2\sqrt{2}$ and $y_{1}=2x-4$
\begin{aligned} &=\int_{0}^{4}(2 \sqrt{x}) d x-\int_{1}^{4}(2 x-4) d x \\ &=\left[\frac{4}{3} x^{\frac{3}{2}}\right]_{0}^{4}-\left[x^{2}-4 x\right]_{1}^{4} \\ \end{aligned}
\begin{aligned} &=\left[\frac{4}{3}(4)^{\frac{3}{2}}\right]-\left[\frac{4}{3}(0)^{\frac{3}{2}}\right]-\left[4^{2}-4(4)\right]-[1-4(1)] \\ \end{aligned}
\begin{aligned} &=\left[\frac{32}{3}-0\right]-[0-(1-4)] \\ &=\frac{32}{3}-3 \\ &=\frac{23}{3} s q \cdot \text { units } \end{aligned}

The RD Sharma class 12th exercise 20.4 consists of a total of 5 questions that are short and easy, covering up the effectively important concepts of this chapter that are profitable for board exams. The concepts covered in the RD Sharma class 12 solution of Area of bounded region exercise 20.4 are-

• The line segment of the curve

• Area of the special bounded region

• Area of the region that is bounded by the curve

• Area of the region that is bounded by a parabola

The RD Sharma class 12th exercise 20.4 Is trusted by thousands of students and teachers as well when it comes to the subject of mathematics. Students who have passed the Board examinations Praise the RD Sharma solutions and recommend to all other students To make the best use of the solution. The RD Sharma class 12th exercise 20.4 Is prepared by experts who are famous in the field of mathematics to ensure that every student gets the best of the solutions of maths and can easily Ace the subject.

The RD Sharma class 12 chapter 20 exercise 20.4 Updated regularly and every year and as always presented with the latest version that is being used in the academic session. It also matches up with the syllabus of NCERT hence any student who is to prepare for the 12th board exams are any public examinations can very easily go through the RD Sharma class 12th exercise 20.4 and make sure that the self practice and evaluate the performance accordingly.

The RD Sharma class 12 solutions chapter 20 exercise 20.4 Can be downloaded from the carrier 360 website free of cost, any student from any corner of the country can own this material by just clicking on one button on the careers 360 website and access the online PDF of the RD Sharma class 12th exercise 20.4 and all they have to do a study hard and make sure to make the best use of the solutions.

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##### 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
##### 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
##### 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
##### 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