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Rom4ik [11]
3 years ago
9

Find the length of the arc of the circular helix with vector equation r(t) = 2 cos t i + 2 sin t j + tk from the point (2, 0, 0)

to the point (2, 0, 2π).
Mathematics
1 answer:
USPshnik [31]3 years ago
4 0

\vec r(t)=2\cos t\,\vec\imath+2\sin t\,\vec\jmath+t\,\vec k

\implies\vec r'(t)=-2\sin t\,\vec\imath+2\cos t\,\vec\jmath+\vec k

\implies\|\vec r'(t)\|=\sqrt{(-2\sin t)^2+(2\cos t)^2+1^2}=\sqrt5

Then the length of the arc is

\displaystyle\int_0^{2\pi}\|\vec r'(t)\|\,\mathrm dt=\sqrt5\int_0^{2\pi}\mathrm dt=2\sqrt5\,\pi

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Answer:

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