Question

${\int }_0^{100}\left\{\sqrt{x}\right\}dx$, where $\{\cdot \}$ denote fraction part of $\left(x\right)$ is

• A. $\frac{1630}{3}$
• B. $\frac{2000}{3}$
• C. $\frac{1000}{3}$
• D. $\frac{1730}{3}$

Answer 1

The correct option is D $\frac{1730}{3}$

$I={\int }_0^{100}\left\{\sqrt{x}\right\}dx$

Substitute $\sqrt{x}=t\Rightarrow x=t^2\Rightarrow dx=2tdt$

$I={\int }_0^{10}\left\{t\right\}2tdt={\int }_0^{10}(t-\left[t\right])2tdt={\int }_0^{10}2t^{2}dt-{\int }_0^{10}\left[t\right]2tdt=2{\left[\frac{t^3}{3}\right]}_0^{10}-90{\left[t^2\right]}_0^1=\frac{1730}{3}$