Question

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

• A. $\frac{9}{4}$
• B. $\frac{45}{16}$
• C. $\frac{27}{8}$
• D. $\frac{63}{16}$

Answer 1

Answer: (C)

$y=\{\begin{array}{ll}3+(x+1)+\left(x-\frac{1}{2}\right),& x<-1\\ 3-(x+1)+\left(x-\frac{1}{2}\right),& -1\le x<\frac{1}{2}\\ 3-(x+1)-\left(x-\frac{1}{2}\right),& \frac{1}{2}\le x\end{array}$
$y=\{\begin{array}{ll}\frac{7}{2}+2x,& x<-1\\ \frac{3}{2},& -1\le x<\frac{1}{2}\\ \frac{5}{2}-2x,& \frac{1}{2}\le x\end{array}$

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.