Mirror mirror on the graph

Calculus Level 5

A mirror in the form of the graph function \(x^{3}-0.5x\) (Shown in red in the GIF) is shone with parallel vertical rays of light from the bottom of the graph. The reflected rays of light form a droplet-shaped shape (shown in green in the GIF). Given that the area of that green shape is \(A\), find \[\left\lfloor { 10 }^{ 8 } \times A \right\rfloor \]

\(\textbf{Details and Assumptions}\)

\(\bullet\) If the equation is of a semi circle (\(\sqrt{1-x^{2}}\)):

You can solve this without any computational methods, but it is however, quite tedious. As such, you can use any computational methods, including Wolfram Alpha.

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