The 22nd Root Problem

Calculus Level 2

S1=188722+188822++201322+201422 S2=188822+188922++201422+201522I=18872015 ⁣x22dx\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 S1S_{1}, S2S_{2}, and II?

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