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Section 7.4 Surfaces of revolution
Objectives
-
Measure the area of a surface formed by revolving a curve around an axis.
-
Relate each thin surface band to its radius and to the curve’s rate of change.
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Use simplified formulas when the revolving curve is expressed as a function.
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Area of a surface of revolution for a parametric curve about the \(x\)-axis:
\begin{equation*}
S=\int_{a}^{b} 2\pii y\sqrt{\left(\dv{x}{t}\right)^{2}+\left(\dv{y}{t}\right)^{2}}\dd{t}
\end{equation*}
-
If revolving a function \(y=f(x)\text{,}\) this simplifies:
\begin{equation*}
S=\int_{a}^{b} 2\pii f(x)\sqrt{1+f'(x)^{2}}\dd{x}
\end{equation*}
