The diagram below shows the graphs of  $y=16-2x$ and $y={x}^{3}-15{x}^{2}+66x-80.$

Find the total area of the shaded regions.

This question has 2 part(s).

Starting with the equation of a semicircle of radius $r$, $y=\sqrt{{r}^{2}-{x}^{2}}$  (where $r>0$ ),  use calculus to prove that the general formula for the volume,$V$, of a sphere of radius $r$ is

      $V=\frac{4}{3}{ \pi r}^{3}$.

This question has 3 part(s).