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Multivariable Calculus

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Surface Area of Helicoid

MVCALC-XHGPNK

A Helicoid is the surface defined by equations:

$$\begin{equation*} \begin{cases} x(r,\theta) = r\cos\theta \\\ y(r,\theta) = r\sin\theta \\\ z(r,\theta) = \theta \end{cases} \end{equation*}$$

...where $\theta \in [0,2\pi]$ and $r\in [0,2].$ Find its area.

A

$\pi(2\sqrt{5}+\ln(2+\sqrt{5}))$

B

$2\pi(\sqrt{5}+\ln(2+\sqrt{5}))$

C

$\sqrt{5}\pi$

D

$\cfrac{\sqrt{5}\pi}{2}+\sqrt{2}$

E

$\cfrac{\pi}{\sqrt{5}}+1$