Finding out the amount of volume through integration

May 2010
A right circular cone has height 4 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 − x^2 . 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?
Mar 2010
So you have to integrate in two sections - the cone and the paraboloid. The equation for the cone is y=2x and y goes from 0 to 4. The equation for the paraboloid is \(\displaystyle y=8-x^2\) with y going from 4 to 8. The element of volume is a cylinder of radius x and height dy, so its volume is \(\displaystyle \pi{x}^2\ dy\). The volume is given by:

\(\displaystyle V=\int\pi{x}^2\ dy=\int_0^4\pi\left(\frac{y}{2}\right)^2\ dy+\int_4^8\pi(8-y)\ dy\)

- Hollywood