Question

Find the value of ${\int }_0^{7\frac{1}{2}}[x-1]dx$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x$

• A. $5/2$
• B. $1$
• C. $5/4$
• D. $17$

Answer 1

The correct option is D $17$

${\int }_0^{7^1{/}_2}\left(\right[x]-1)dx.$

${\int }_0^{15/2}\left(\right[x]-1)dx={\int }_0^1(-1)dx+{\int }_2^{1}0dx+{\int }_2^{3}1dx+{\int }_3^{4}2dx+{\int }_4^{5}3dx+{\int }_5^{6}4dx+{\int }_6^{7}5dx+{\int }_7^{7.5}6dx.$

$=-1+0+1+2+3+4+5+6\left(0.5\right)$

$=17$