Question

The area bounded by the graph $y=\left|\left[x-3\right]\right|$ , above the $x$ -axis from $x=-2$ to $x=3$ is ( $\left[.\right]$ denotes the greatest integer function)

• A. $7sq.units$
• B. $15sq.units$
• C. $21sq.units$
• D. $28sq.units$

Answer 1

Answer: (B)

$\therefore$ Required area
$={\int }_{-2}^3\left|\left[x-3\right]\right|dx$
$={\int }_{-2}^{-1}\left|\left[x-3\right]\right|dx+{\int }_{-1}^0\left|\left[x-3\right]\right|dx$
$+{\int }_0^1\left|\left[x-3\right]\right|dx+{\int }_1^2\left|\left[x-3\right]\right|dx+{\int }_2^3\left|\left[x-3\right]\right|dx$
$={\int }_{-2}^{-1}5dx+{\int }_{-1}^{0}4.dx+{\int }_0^{1}3.dx+{\int }_1^{2}2.dx+{\int }_2^{3}1.dx$
$=5\left(1\right)+4\left(1\right)+3\left(1\right)+2\left(1\right)+1\left(1\right)$
$=15sq.units$ .