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## 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.

## Solution

Consider half a parabola where the interval of is :

where is constant.

At , we have .

At , we have .

Solving for the constant yields ; thus,

.

Use the disk method to calculate the volume:

When compared with the volume the cylinder , the volume of the paraboloid is half the volume of the cylinder.

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