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If we have a curve lying entirely on or above the x-axis, which is the graph of a function f(x), and x is restricted to being between a and b, then we may rotate it about the x-axis, generating a three dimensional object; a solid of revolution.

For example, if the curve is simply f(x) = ax/h (a straight line) and x is between 0 and h, we obtain a cone of height h and radius a.

If tex2html_wrap_inline165 and x is between, say, 0 and 1, we obtain a bowl shaped object called a paraboloid. (Note the equation tex2html_wrap_inline167 is equivalent to tex2html_wrap_inline169 , a parabola).

If tex2html_wrap_inline171 and x is between -a and a, we obtain a sphere of radius a (since the equation tex2html_wrap_inline179 is equivalent to tex2html_wrap_inline181 , a circle of radius a).

The textbook (p. 388) develops the formula that the volume of the solid obtained by rotating about the x-axis is given by


The volume is obtained by adding up volumes of cylindrical disks, while its surface area (p. 400) is given by


Note that both of these formulas are for rotating the graph of f(x) about the x-axis.

Sean O Anderson
Tue Dec 3 12:06:23 EST 1996