## Solids or Volumes of Revolution

We start with a graph If the graph is rotated about the axis it traces out a surfaces as shown. Between the surface and the – axis we may form a solid. We show here how to find the volume of this solid. We may picture the solid as being made up of slices of solid. For the function each slice is a disk of radius and thickness and has volume By summing these slices, obtaining we get an approximate value for the volume. The value becomes exact turning the summation into an integral. Hence, if a curve between the values of and is rotated about the - axis, the volume of the solid formed is (1)

If we have a curve which we rotate about the - axis between and the volume of the solid formed is (2) obtained from (1) by interchanging and Example: The curve is rotated about the – axis. Find the volume of the solid formed. We can integrate by using the identity to give Example: The graph is rotated about the - axis. Find the volume of the solid formed. We evaluate:  