Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Please solve RD Sharma Class 12 Chapter 20 Areas of Bounded Regions Exercise 20.3 Question 43 Maths textbook solution.

Please solve RD Sharma Class 12 Chapter 20 Areas of Bounded Regions Exercise 20.3 Question 43 Maths textbook solution.

Answers (1)

Answer:

\pi -2  sq. units

Hints:

Use concept.

Given:

\left\{(x, y): x^{2}+y^{2} \leq 4, x+y \geq 2\right\}

Solution: The equation of the given curves are

x^2+y^2=4 ……(1)

x+y=2  …….(2)

Clearly x^2+y^2=4 represents a circle and x+y=2 is the equation of a straight line cutting x and y axis at (0,2) and (2,0) respectively.

The smaller region bounded by thesetwo curves is shaded in the following figure.

Length = y_2-y_1

Width = x and

Area = (y_2-y_1)x

Since the approximating rectangle can move from x=0 to x=2, the required area is given by

\begin{aligned} &\mathrm{A}=\int_{0}^{2}\left(y_{2}-y_{1}\right) d x\\ &\text { We have } y_{1}=2-x \text { and } y_{2}=\sqrt{4-x^{2}}\\ &\text { Thus, } A=\int_{0}^{2}\left(\sqrt{4-x^{2}}-2+x\right) d x\\ &\Rightarrow \mathrm{A}=\int_{0}^{2}\left(\sqrt{4-x^{2}}\right) d x-2 \int_{0}^{2} d x+\int_{0}^{2} x d x \end{aligned}

\begin{aligned} &\Rightarrow A=\left[\frac{x \sqrt{4-x^{2}}}{2}+\frac{a^{2}}{2} \sin ^{-1}\left(\frac{x}{2}\right)\right]_{0}^{2}-2(x)_{0}^{2}+\left(\frac{x^{2}}{2}\right)_{0}^{2} \\ &\Rightarrow A=\frac{4}{2} \sin ^{-1}\left(\frac{2}{2}\right)-4+2 \\ &\Rightarrow A=2 \sin ^{-1}(1)-2 \\ &\Rightarrow A=2 \times \frac{\pi}{2}-2 \\ &\Rightarrow A=\pi-2 \end{aligned}

Therefore, the area of the region is \pi -2  sq. units

Posted by

infoexpert24

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Date Sheet 2025 (Out) Live: Class 10, 12 board exam time table at cbse.gov.in; syllabus, sample papers
anam-khan

CBSE Class 12 board exam date sheet 2025 out; exams from February 15 to April 4
anam-khan

When will CBSE Board exam 2025 date sheet be released for Class 10, 12?

Latest Question