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.

`{: ("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"`.