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  4. The area of the bounded region enclosed by the curve y=3-|x-(1/2)|-|x+1| and the x-axis is

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Q. The area of the bounded region enclosed by the curve $y=3-\left|x-\frac{1}{2}\right|-|x+1|$ and the $x$-axis is

JEE MainJEE Main 2022Application of Integrals

Solution:

$y= \begin{cases}3+(x+1)+\left(x-\frac{1}{2}\right), & x<-1 \\ 3-(x+1)+\left(x-\frac{1}{2}\right), & -1 \leq x<\frac{1}{2} \\ 3-(x+1)-\left(x-\frac{1}{2}\right), & \frac{1}{2} \leq x\end{cases}$
$y = \begin{cases}\frac{7}{2}+2 x , & x <-1 \\ \frac{3}{2}, & -1 \leq x <\frac{1}{2} \\ \frac{5}{2}-2 x , & \frac{1}{2} \leq x \end{cases}$
image
Area bounded $=$ ar $ABF +$ ar $BCEF +$ ar $CDE$
$=\frac{1}{2}\left(\frac{3}{4}\right)\left(\frac{3}{2}\right)+\left(\frac{3}{2}\right)\left(\frac{3}{2}\right)+\frac{1}{2}\left(\frac{3}{4}\right)\left(\frac{3}{2}\right)$
$=\frac{27}{8} sq$. units.