Inspired by Michael Mendrin

Calculus Level 5

A rectangle's base lie on the $$x$$-axis while it's top corners are on the curve $$\displaystyle y =\dfrac{x}{x^2 + 1}$$. Now a cylinder is obtained by revolving Rectangle about x-axis, if the maximum volume of cylinder is $$V_{max}$$. Find $$\displaystyle \lfloor 1000V_{max} \rfloor$$.

Inspired by this awesome problem

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