Get Answers to all your Questions

header-bg qa

Please solve RD Sharma Class 12 Chapter 20 Areas of Bounded Regions Exercise 20.3 Question 46 Maths textbook solution.

Answers (1)

Answer:

\frac{1}{2} .sq. Units.

Hints:

Use concept.

Given:

The curves y=\left | x-1 \right |  and y=-\left | x-1 \right |+1

Solution:

To find area enclosed by

\begin{aligned} &y=|x-1| \\ &\Rightarrow y=\left\{\begin{array}{l} -(x-1), \text { if } x-1<0 \\ (x-1), \text { if } x-1 \geq 0 \end{array}\right. \end{aligned}

\begin{aligned} &\Rightarrow y=\left\{\begin{array}{l} 1-x, \text { if } x<1 \\ x-1, \text { if } x \geq 1 \end{array}\right. \\ &\text { And } y=-|x-1|+1 \\ &\Rightarrow y=\left\{\begin{array}{l} +(x-1)+1, \text { if } x-1<0 \\ -(x-1)+1, \text { if } x-1 \geq 0 \end{array}\right. \\ &y=\left\{\begin{array}{lr} x, \quad \text { if } x<1 \\ -x+2, \text { if } x \geq 1 \end{array}\right. \end{aligned}

A rough sketch of equation of lines (1),(2),(3),(4) is given as:

Shaded region is the required area.

Required area = Region ABCDA

Required area = Region BDCB + Region ABDA

Region BDCB is sliced into rectangles of area  =(y_2-y_1)x and it slides from x=\frac{1}{2}  to x =1

Region ABDA is sliced into rectangle of area =(y_3-y_4)x and it slides from x =1 to x=\frac{3}{2} .

So, using equation (1),

Required area = Region BDCB + Region ABDA

\begin{aligned} &=\int_{\frac{1}{2}}^{1}\left(y_{1}-y_{2}\right) d x+\int_{1}^{\frac{3}{2}}\left(y_{3}-y_{4}\right) d x \\ &=\int_{\frac{1}{2}}^{1}(x-1+x) d x+\int_{1}^{\frac{3}{2}}(-x+2-x+1) d x \\ &=\int_{\frac{1}{2}}^{1}(2 x-1) d x+\int_{1}^{\frac{3}{2}}(3-2 x) d x \\ &=\left[x^{2}-x\right]_{\frac{1}{2}}^{1}+\left[3 x-x^{2}\right]_{1}^{\frac{3}{2}} \end{aligned}

\begin{aligned} &=\left[(1-1)-\left(\frac{1}{4}-\frac{1}{2}\right)\right]+\left[\left(\frac{9}{2}-\frac{9}{4}\right)-(3-1)\right] \\ &=\frac{1}{4}+\frac{9}{4}-2 \\ &A=\frac{1}{2} \text { sq. units } \end{aligned}

Posted by

infoexpert24

View full answer

Crack CUET with india's "Best Teachers"

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