Mirror mirror on the graph

Calculus Level 5

LATEX

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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