Problem

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

Paraboloid.jpg

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.

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