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 44 Maths textbook solution.

Please solve RD Sharma Class 12 Chapter 20 Areas of Bounded Regions Exercise 20.3 Question 44 Maths textbook solution.

Answers (1)

Answer: \left(\frac{3 \pi}{2}-3\right) \text { sq. units }

Hints:

Use concept.

Given:

\left\{(x, y): \frac{x^{2}}{9}+\frac{y^{2}}{4} \leq 1 \leq \frac{x}{3}+\frac{y}{2}\right\}

Solution:

To find the area of a region

\left\{(x, y): \frac{x^{2}}{9}+\frac{y^{2}}{4} \leq 1 \leq \frac{x}{3}+\frac{y}{2}\right\}

Here,

\frac{x^2}{9}+\frac{y^2}{4}=1    …..(1)

\frac{x}{3}+\frac{y}{2}=1        …..(2)

Equation (1) represents an ellipse with centre at origin and meets axis at (\pm 3,0) ,(0,\pm 2) .

Equation (2) is a line that meets axis at (3,0) ,(0,2)

A rough sketch is as under:

Shaded region represents required area. This is sliced into rectangles with area (y_2-y_2)x  which slides from x=0  to  x=3, so

Required area = Region APBQA

\begin{aligned} &A=\int_{0}^{3}\left(y_{1}-y_{2}\right) d x \\ &=\int_{0}^{3}\left[\frac{2}{3} \sqrt{9-x^{2}} d x-\frac{2}{3}(3-x) d x\right] \\ &=\frac{2}{3}\left[\frac{x}{2} \sqrt{9-x^{2}}+\frac{9}{2} \sin ^{-1}\left(\frac{x}{3}\right)-3 x+\frac{x^{2}}{2}\right]_{0}^{3} \\ &=\frac{2}{3}\left[\left\{0+\frac{9}{2} \cdot \frac{\pi}{2}-9+\frac{9}{2}\right\}-\{0\}\right] \end{aligned}

\begin{aligned} &=\frac{2}{3}\left[\frac{9 \pi}{4}-\frac{9}{2}\right] \\ &A=\left(\frac{3 \pi}{2}-3\right) \text { sq. units } \end{aligned}

Therefore, the area of the region is \left(\frac{3 \pi}{2}-3\right) \text { 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

Board Exam Time Table 2025 Live: UP Board exams from February 24; CBSE, BSEB date sheet soon
anam-khan

CBSE Date Sheet 2025 Live: Class 10, 12 board exam dates soon; syllabus, sample papers
anam-khan

Extra time in CBSE exams for students with type 1 diabetes: Kerala SHRC seeks report on plea

Latest Question