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Solids or Volumes of Revolution

We start with a graphIf the graph is rotated about theaxis 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 functioneach slice is a disk of radiusand thicknessand has volumeBy summing these slices, obtainingwe get an approximate value for the volume. The value becomes exactturning the summation into an integral. Hence, if a curve between the values ofandis rotated about the- axis, the volume of the solid formed is(1)

If we have a curvewhich we rotate about the- axis betweenandthe volume of the solid formed is(2) obtained from (1) by interchangingand

Example: The curveis rotated about the– axis. Find the volume of the solid formed.

We can integrate by using the identityto give

Example: The graphis rotated about the- axis. Find the volume of the solid formed.

We evaluate: