Arc length of r(t) = (cos t, sin t, t) for t in [0, 2π] is which of the following?

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Multiple Choice

Arc length of r(t) = (cos t, sin t, t) for t in [0, 2π] is which of the following?

Arc length is found by integrating the speed of the curve: length = ∫ from 0 to 2π of ||r'(t)|| dt. Compute r'(t) for r(t) = (cos t, sin t, t): r'(t) = (-sin t, cos t, 1). The speed is the norm of this derivative: ||r'(t)|| = sqrt[(-sin t)^2 + (cos t)^2 + 1^2] = sqrt[sin^2 t + cos^2 t + 1] = sqrt[1 + 1] = sqrt(2). Since the speed is constant, the arc length is ∫_0^{2π} sqrt(2) dt = sqrt(2) · (2π) = 2π√2. So the length is 2π√2.

Arc length is found by integrating the speed of the curve: length = ∫ from 0 to 2π of ||r'(t)|| dt.

Compute r'(t) for r(t) = (cos t, sin t, t): r'(t) = (-sin t, cos t, 1). The speed is the norm of this derivative:

||r'(t)|| = sqrt[(-sin t)^2 + (cos t)^2 + 1^2] = sqrt[sin^2 t + cos^2 t + 1] = sqrt[1 + 1] = sqrt(2).

Since the speed is constant, the arc length is ∫_0^{2π} sqrt(2) dt = sqrt(2) · (2π) = 2π√2.

So the length is 2π√2.

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