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Volume and Surface Area of Spheres

The volume of a sphere of radius r is tex2html_wrap_inline663 .

The surface area of a sphere of radius r is tex2html_wrap_inline667 .

Why is the formula for the surface area S equal to the derivative of the formula for the volume? Because when r is increased very slightly by an amount tex2html_wrap_inline645 , the volume V of the sphere increases very slightly by an amount tex2html_wrap_inline677 approximately equal to tex2html_wrap_inline679 (a ``shell'' around the sphere of area S and width tex2html_wrap_inline645 ). So tex2html_wrap_inline685 , and in the limit dV/dr=S.

Carl Lee
Wed Apr 21 08:17:28 EDT 1999