Moving Dot P on a Plane

Calculus Level pending

The coordinates of a dot P P moving on the x-y plane at time t t is given by: x = ∫ 0 t θ cos ⁡ θ d θ and y = ∫ 0 t θ sin ⁡ θ d θ . x = \int_0^t\theta \cos\theta\ d\theta \text{ and } y = \int_0^t\theta \sin \theta\ d\theta. What is the distance traveled by P P in the interval 0 ≤ t ≤ 20 0 \leq t \leq 20 ?


The answer is 200.

Arron Kau by Arron Kau

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1 solution

Arron Kau Staff
May 13, 2014

Let L L be the distance, so we have

L = ∫ 0 20 ( d x d t ) 2 + ( d y d t ) 2 d t = ∫ 0 20 ( t cos ⁡ t ) 2 + ( t sin ⁡ t ) 2 d t = ∫ 0 20 t 2 ( cos ⁡ 2 t + sin ⁡ 2 t ) d t = ∫ 0 20 t d t = [ t 2 2 ] 0 20 = 200 \begin{aligned} L &= \int_0^{20}\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\ dt \\ &= \int_0^{20}\sqrt{(t\cos t)^2 + (t \sin t)^2}\ dt \\ &= \int_0^{20} \sqrt{t^2(\cos^2 t + \sin^2 t)}\ dt \\ &= \int_0^{20} t\ dt \\ &= \left[\frac{t^2}{2}\right]_0^{20} \\ &= 200 \\ \end{aligned}

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