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Home » Setting the volume equal to \( 288\pi \): - Crosslake

Setting the volume equal to \( 288\pi \): - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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We now find the maximum of $f(u)$ on this interval. Note that as $u \t...
But wait: as $u \to 0^+$, $\frac{1}{u^2} \to \infty$, so $f(u) \to \in...
Let’s test at $x = \frac{\pi}{4}$: $\sin x = \cos x = \frac{\sqrt{2}}{...
So $u = \frac{1}{2}$ at $x = \frac{\pi}{4}$, and $f\left(\frac{1}{2}\r...
Now check the behavior as $u \to 0^+$:
\frac{1}{u^2} \to \infty, \quad \frac{2}{u} \to \infty, \quad 2u \to 0
So $f(u) \to \infty$
But this contradicts intuition — is the expression unbounded?
Wait: let’s recompute at small $x$, say $x = 0.1$ radians:
$\sin x \approx 0.0998$, $\cos x \approx 0.995$, so $\sec x \approx 1....

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