Table 5.5.1 summarizes the contents of the "formula" that gives the surface area of a surface of revolution generated by rotating a curve about either a horizontal axis () or a vertical axis (. The distance of an arc-length element from the axis of rotation is , so for curves defined either explicitly or parametrically, Table 5.5.1 lists the appropriate expressions for and .
Curve
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Axis of Rotation
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Table 5.5.1 Surface area of a surface of revolution:
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The arc-length element appears in the surface-area integrals because the surface area is defined as the limiting sum of the areas of approximating frustums of a cone. (A frustum of a cone is a segment cut off by two planes parallel to the base.) If and are respectively the smaller and larger radii of a frustum whose generator (lateral height) has length , then the surface area of the frustum is . As the number of frustums increases, is replaced by , and by . Hence, the surface-area element becomes , and the integrals in Table 5.5.1 follow.