By: Tao Steven Zheng (郑涛)

【Problem】

Use integration to derive the volume of a paraboloid with a base radius and height . Compare the volume of the paraboloid to the volume of the cylinder with equal base and height.

【Solution】

Consider half a parabola where the interval of is .

At , ; at , .

Therefore,

and

Use the disk method to calculate the volume.

Comparing with the volume the cylinder, , the volume of the paraboloid is half the volume of the cylinder.