The 22nd Root Problem

Calculus Level 2

$\begin{aligned} S_{1}&=&\sqrt[22]{1887}+\sqrt[22]{1888}+\cdots+\sqrt[22]{2013}+\sqrt[22]{2014} \\ \\ \ S_{2}&=&\sqrt[22]{1888}+\sqrt[22]{1889}+\cdots+\sqrt[22]{2014}+\sqrt[22]{2015} \\ \\ I&=&\int_{1887}^{2015}\!\sqrt[22]{x}\,\mathrm{d}x \end{aligned}$

What can you say about the relative values of $S_{1}$ , $S_{2}$ , and $I$ ?

S_{2}<S_{1}<I

I<S_{1}<S_{2}

S_{1}<I<S_{2}

S_{2}<I<S_{1}

I<S_{2}<S_{1}

S_{1}<S_{2}<I