Area of Surface of Revolution from Astroid
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Theorem
Let $H$ be the astroid constructed within a circle of radius $a$.
The surface of revolution formed by rotating $H$ around the $x$-axis:
evaluates to:
- $\SS = \dfrac {12 \pi a^2} 5$
Proof
By symmetry, it is sufficient to calculate the surface of revolution of $H$ for $0 \le x \le a$.
From Area of Surface of Revolution, this surface of revolution is given by:
- $\ds \SS = 2 \int_0^{\pi / 2} 2 \pi y \, \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$
From Equation of Astroid:
- $\begin{cases}
x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$
so:
\(\ds \frac {\d x} {\d \theta}\) | \(=\) | \(\ds -3 a \cos^2 \theta \sin \theta\) | ||||||||||||
\(\ds \frac {\d y} {\d \theta}\) | \(=\) | \(\ds 3 a \sin^2 \theta \cos \theta\) |
Hence:
\(\ds \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2}\) | \(=\) | \(\ds \sqrt {9 a^2 \paren {\sin^4 \theta \cos^2 \theta + \cos^4 \theta \sin^2 \theta} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 a \sqrt {\sin^2 \theta \cos^2 \theta \paren {\sin^2 \theta + \cos^2 \theta} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 a \sqrt {\sin^2 \theta \cos^2 \theta}\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 a \sin \theta \cos \theta\) |
Thus:
\(\ds \SS\) | \(=\) | \(\ds 2 \int_0^{\pi / 2} 2 \pi a \sin^3 \theta \, 3 a \sin \theta \cos \theta \rd \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 12 \pi a^2 \int_0^{\pi / 2} \sin^4 \theta \cos \theta \rd \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 12 \pi a^2 \intlimits {\frac {\sin^5 \theta} 5} 0 {\pi / 2}\) | Primitive of $\sin^n a x \cos a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {12 \pi a^2} 5 \paren {\sin^5 \theta \frac \pi 2 - \sin^5 \theta 0}\) | evaluating limits of integration | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {12 \pi a^2} 5 \paren {1 - 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {12 \pi a^2} 5\) |
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Problem $8$