Circular cone volume through integration

[orangesun]

Homework Statement

A right circular cone has height 6 cm and base radius 2. It is over-filled with ice cream,
in the usual way. Place the cone so its vertex is at the origin, and its axis lies along the
positive y–axis, and take the cross-section containing the x–axis. The top of this crosssection is a piece of the parabola y = 8 − x² . The whole filled ice-cream cone is obtained
by rotating this cross-section about the y–axis.
What is the volume of the ice cream?

Homework Equations

y = 8 − x²

The Attempt at a Solution

I have no idea how to work out this question,
but I think where you start off is by integrating y = 8 − x²
so that ends up y = 8x - x³/3

I just don't know how to get to the next step from here.

Many thanks.

 

[LCKurtz]

Homework Statement

A right circular cone has height 6 cm and base radius 2. It is over-filled with ice cream,
in the usual way. Place the cone so its vertex is at the origin, and its axis lies along the
positive y–axis, and take the cross-section containing the x–axis. The top of this crosssection is a piece of the parabola y = 8 − x² . The whole filled ice-cream cone is obtained
by rotating this cross-section about the y–axis.
What is the volume of the ice cream?

Homework Equations

y = 8 − x²

The Attempt at a Solution

I have no idea how to work out this question,
but I think where you start off is by integrating y = 8 − x²
so that ends up y = 8x - x³/3

I just don't know how to get to the next step from here.

Many thanks.

I think you have something typed wrong because the parabola doesn't intersect the cone at its rim (it should go through the point (2,6). Once that is fixed, use the method of shells to calculate the revolved volume.