To illustrate the fact that the numbers we get for length do not depend on the way we parametrize our curves (except for the mild restrictions preventing doubling back mentioned earlier), calculate the length of the semicircle y = √1 - x² with these two different parametrizations:

a. x = cos 2t, y = sin 2t, 0 ≤ t ≤ π/2.

b. x = sin πt, y = cos πt, -1/2 ≤ t ≤ 1/2.