The shaded region in the picture lies between the $y$-axis and the curve $y = 2x^2$, from $y = 0$ up to $y = 8$. It is turned about the $y$-axis to make a bowl, with the lengths in centimetres. What is the volume of the bowl?

Fig. 1: A graph with the y-axis vertical. The curve y = 2x squared rises from the origin to the height 8 at x = 2. A dashed horizontal line marks y = 8. The region between the y-axis and the curve, from y = 0 to y = 8, is shaded light blue.