Volume of a paraboloid (Archimedes) The region bounded by the parabola \(y = ax^2\) and the horizontal line y = h is revolved about the y-axis to generate a solid bounded by a surface called a paraboloid (where a > 0 and h > 0). Show that the volume of the solid is \(\frac{3}{2}\) the volume of the cone with the same base and vertex.

Solution 82E

Step 1 of 2:

In this problem we need to show that the volume of the solid is the volume of the cone with same base and vertex.

First we need to find the volume of the solid that is revolved about the y-axis.

Given : The region bounded by the parabola y = a, and the horizontal line y = h is revolved about the y-axis.

The related graph of The region bounded by the parabola y = a, and the horizontal line y = h is shown below :