 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2021-resit/4/solution

\overset{x}{˙} = a t cos t, \overset{y}{˙} = a t sin t .

The arc length element is

d s = \overset{x}{˙}^{2} + \overset{y}{˙}^{2} d t = ∣ a ∣ t d t

0 \leq t \leq 2

. Thus

L = \int_{0}^{2} ∣ a ∣ t d t = 2∣ a ∣.

For a positive geometric scale

a

, this is

2 a

. The absolute value is necessary when the unspecified real constant

a

is negative;

a = 0

gives zero arc length.

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