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

Answer 1

Answer: (B)

$∴ \, $ Required area
$=\displaystyle \int _{- 2}^{3} \left|\left[x - \, 3\right]\right|dx$
$=\displaystyle \int _{- 2}^{- 1} \left|\left[x - \, 3\right]\right|dx \, + \, \displaystyle \int _{- 1}^{0} \left|\left[x - \, 3\right]\right|dx$
$+ \, \displaystyle \int _{0}^{1} \left|\left[x - \, 3\right]\right|dx \, + \, \displaystyle \int _{1}^{2} \left|\left[x - \, 3\right]\right|dx \, + \, \displaystyle \int _{2}^{3} \left|\left[x - \, 3\right]\right|dx$
$=\displaystyle \int _{- 2}^{- 1} 5dx \, + \, \displaystyle \int _{- 1}^{0} 4. \, dx \, + \, \displaystyle \int _{0}^{1} 3 . d x \, + \, \displaystyle \int _{1}^{2} 2 . d x+ \, \displaystyle \int _{2}^{3} 1 . d x$
$=5\left(1\right) \, + \, 4\left(1\right) \, + \, 3\left(1\right) \, + \, 2\left(1\right) \, +1\left(1\right)$
$=15 \, sq. \, units$ .