What is the length of the curve with parametric equations x = t - sin(t), y = 1 - cos(t)

Question

What is the length of the curve with parametric equations x = t - sin(t), y = 1 - cos(t) from t = 0 to

t = 2π? (4 points)

answer choices

	8
	4
	8π
	2π

Answer 1

dx/dt = 1 - cos(t)

dy/dt = sin(t)

We use I[a, b] for the integral from a to b and E[a,b] for the evaluation from a to b.

I[0, 2π] √(1 - cos(t))² + (sin(t))²) dt =

I[0, 2π] √(1 - 2 cos(t)+ cos²(t) + sin²(t)) dt =

I[0, 2π] √(1 - 2 cos(t)+ cos²(t) + sin²(t)) dt =

I[0, 2π] √(2 - 2 cos(t)) dt = (recalling that √(1 - cos(t))/2)

I[0, 2π] 2 |sin t/2| dt = ,as sin(t/2) >= 0 on [0, 2π]

I[0, 2π] 2 sin t/2| dt =

-4 cos t/2 E[0,2π] =

4 --4 = 8