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Solids of Revolution

Integrals are used in this section to find the volume of a solid of revolution. A solid of revolution is generated by taking a region in the first quadrant of the plane and rotating it in space about the x- or y-axis (Figure 6.2.1).


Figure 6.2.1 Solids of Revolution

We shall work with the region under a curve and the region between two curves. We use one method for rotating about the axis of the independent variable and another for rotating about the axis of the dependent variable.

For areas our starting point was the formula

area = base × height

for the area of a rectangle. For volumes of a solid of revolution our starting point is the usual formula for the volume of a right circular cylinder (Figure 6.2.2).


Figure 6.2.2


The volume of a right circular cylinder with height h and base of radius r is

V = πr2h.

Last Update: 2006-11-22