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=-(x^2)/(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.

We now apply the formula for the volume of a solid of revolution:

"Vol"=pi int_a^by^2\ dx

=pi int_-50^50(-(x^2)/(250)+40)^2 dx

=pi int_-50^50((x^4)/(62500) - (80x^2)/(250)+1600) dx

=pi[(x^5)/(312500)-(80x^3)/(750)+1600x]_-50^50

Now, since

(−50)5 = −505,

(−50)3 = −503, and

(−50) = −50,

we can reduce the amount of writing somewhat and put:

text[Vol] = 2pi[((50)^5)/(312500)-(80(50)^3)/(750)+1600(50)]

=425162\ text[cm]^3

=425.2\ text[L]

So the wine cask will hold 425.2\ "L".