In this video I go over another example on determining the arc length of a parametric curve, and this time revisit the unit circle but instead deal with a slightly different set of parametric equations, namely x = sin(2t) and y = cos(2t) for the interval where t is between 0 and 2pi. The interesting thing about this curve is that the point traced on the curve completes two full cycles or revolutions of the circle. This means that using the arc length formula measures double the parametric curve, or twice circumference of the circle, namely 4pi. This is a very important example because it illustrates the need to always be careful about selecting the interval of the parametric curve to ensure that the curve is only traced once, otherwise the arc length will be longer than expected, so make sure to watch this video!
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Parametric Curves: Example 2: Unit Circle:
Trigonometry Identities: Proof that sin^2(x) + cos^2(x) = 1:
Parametric Curves: Example 3: Unit Circle: .
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I don't always measure the arc length of a parametric circle but when I do I usually measure the length twice ;)
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/parametric-calculus-arc-length-example-2-unit-circle