Find the surface area generated by rotating the astroid

Find the surface area generated by rotating the astroid $x^{2/3}+y^{2/3}=a^{2/3}$ about the $x$-axis.

Step 1: Parametrize and set up

From Problem 92: $x=a\cos^3t,\ y=a\sin^3t$, with speed $\sqrt{(dx/dt)^2+(dy/dt)^2}=3a\cos t\sin t$ for $t\in[0,\pi/2]$.

Step 2: Integral over the upper quarter

\[S_{\text{upper}}=2\pi\int_0^{\pi/2}a\sin^3t\cdot3a\cos t\sin t\,dt = 6\pi a^2\int_0^{\pi/2}\sin^4t\cos t\,dt\]

Step 3: Evaluate

With $u=\sin t$: \[S_{\text{upper}}=6\pi a^2\int_0^1u^4\,du = 6\pi a^2\cdot\frac15=\frac{6\pi a^2}5\]

Step 4: Double for the lower half

\[S=2\times\frac{6\pi a^2}5 = \frac{12\pi a^2}5\]

\[\boxed{S=\frac{12\pi a^2}5\approx7.54a^2}\]

Step 5: Comparison

A sphere of radius $a$ has surface area $4\pi a^2$; the astroid's surface is $\frac{12/5}{4}=\frac35$ of that — a reflection of the astroid's concave, pinched profile compared to a circle.