The circumference of the unit circle is 2π, so we know after evaluating the integral we should get 2π. This problem didn't tell us values for *t*, which means we have to figure them out ourselves. These equations trace the unit circle once if we take 0 ≤ *t* ≤ π. We have So the length of the curve is Since sin^{2}(2*t*) + cos^{2}(2*t*) = 1, we can evaluate this integral by hand. Boom. This is the right answer. In the example, if we had picked 0 ≤ *t* ≤ 2π, we wouldn't have gotten the right answer. It's important to pick the correct values for *t* so that the part of the curve we're interested in gets traced exactly once. |