Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation: $b^2{x}^2+b^2{y}^2+a^2{z}^2=a^2{b}^2$.

Phil wants to know how much chocolate is needed to cover one candy centre with a uniform coat of chocolate one millimeter thick.

If $a=1$ mm and $b=1$ mm, the amount of chocolate required is ${\displaystyle \frac{28}{3}}\pi$ mm³

If $a=2$ mm and $b=1$ mm, the amount of chocolate required is approximately 60.35475635 mm³.

Find the amount of chocolate in mm³ required if $a=3$ mm and $b=1$ mm. Give your answer as the number rounded to 8 decimal places behind the decimal point.