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

The Attempt at a Solution

So for I have
x² = 8-y
v = pi . integral((8-y)dx) from 0 to 8

I am not sure if I am on the right path though.
Many thanks,

 

[tiny-tim]

Welcome to PF!

Hi orangesun! Welcome to PF!

(have an integral: ∫ and a pi: π )

So for I have
x² = 8-y
v = pi . integral((8-y)dx) from 0 to 8

I am not sure if I am on the right path though.

Yes, that's the right path for the curved part of the ice-cream.

(except it isn't dx, it's dy … each horizontal slice is a disc of area πx² and height dy)

Now you need to decide on the limits of integration (for y), and then add the volume of the cone part.

(btw, is your parabola correct? it doesn't seem to meet the top of the cone … and we wouldn't want to lose any ice-cream! )