Home / A: $\int_0^{4\pi} \sqrt{\sin^2 t + \cos^2 t + \left(\frac{1}{4\pi}\right)^2} \, dt = 4\pi \sqrt{1 + \frac{1}{16\pi^2}}$

A: $\int_0^{4\pi} \sqrt{\sin^2 t + \cos^2 t + \left(\frac{1}{4\pi}\right)^2} \, dt = 4\pi \sqrt{1 + \frac{1}{16\pi^2}}$

February 22, 2026

$A: $\int_0^{4\pi} \sqrt{\sin^2 t + \cos^2 t + \left(\frac{1}{4\pi}\right)^2} \, dt = 4\pi \sqrt{1 + \frac{1}{16\pi^2}}$$

D: $4(2) + 3(1) + 5(-1) = 8 + 3 - 5 = 6$ kJ
Answer: C
Question: A geologist studying cave formations observes that a stalactite grows in a spiral path modeled by the parametric equations $x(t) = \cos t$, $y(t) = \sin t$, $z(t) = \frac{t}{4\pi}$, where $t \geq 0$. Find the arc length of the stalactite's growth from $t = 0$ to $t = 4\pi$ years.
B: $4\pi$ meters
C: $2\pi\sqrt{1 + \frac{1}{16\pi^2}}$
D: $4\pi \sqrt{1 + \frac{1}{16\pi^2}}$