# Volumes of revolution.

Proposition 8.7.1   Let be a continuous function on the interval , with . Suppose that for any , . The volume of the solid of revolution obtained by revolving about the axis the region between the axis and the graph of id equal to:

Example 8.7.2   Revolve about the axis the arc of parabola defined by (v.s. Fig. 6.4). The volume of the horn'' is given by:

If the region to be revolved is not bordered by the axis, the section of the solid is not a disk, but an annulus.

Denote by the function whose graph is the outer boundary of the region , and by the function whose graph is the inner boundary of ; suppose that both functions are continuous on , where .

Proposition 8.7.3   The volume of the solid obtained by revolving the region about the axis is given by:

Example 8.7.4

We revolve about the axis the region bounded by the parabolas whose respective equations are and , by the axis and the line whose equation is , i.e. the region in Figure  7.

The volume of the solid is given by:

Noah Dana-Picard 2007-12-28