Finding volumes of revolution

You are given a curve y=f(x), and you are asked to find the volume of the solid generated when this curve is rotated about the x or y axis.

It is similar to finding the area under the curve. To find the area you divide the shape up into infinitesimally thin strips, each of length f(x) and thickness dx, so the area of each strip is f(x) dx and the total area is the integral of this.

Finding the volume of revolution is similar, but now you are dividing the shape up into infinitesimally thin disks. The details depend on which axis the shape is revolved around.

The x-axis

In this case each disk is perpendicular to the x axis and the radius of the disk is the y value at that point - i.e. f(x). The thickness of the disk is dx.

The volume of the disk is therefore π(f(x))²dx, and the volume of the entire shape is -mathformula not MarkupType-
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