Question

The area bounded by the graph $y=\left|[x-3]\right|$, the $x-$axis and the lines $x=-2$ and $x=3$ is ($[.]$ denotes the greatest Integer function):

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

Answer 1

The correct option is B $15$ sq.unit

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