We will lay the cask on its side to make the algebra easier:

We need to find the equation of a parabola with vertex at (0,40) and passing through (50,30).

We use the formula:

(x − h)² = 4a(y − k)

Now (h, k) is (0, 40) so we have: x² = 4a(y − 40) and the parabola passes through (50, 30), so

(50)² = 4a(30 − 40)

2500 = 4a(-10) and this gives 4a = -250

So the equation of the side of the barrel is

x² = -250(y − 40), that is,

y = -x²/250 + 40

We need to find the volume of the cask which is generated when we rotate this parabola between x = -50 and x = 50 around the x-axis.

Now, since

(-x)⁵ = -x⁵,

(-x)³ = -x³,

etc,

we can write:

So the wine cask will hold 425.2 L.