Answer:
L = 13.3649
Step-by-step explanation:
We are given;
x = t − 2 sin(t)
dx/dt = 1 - 2 cos(t)
Also, y = 1 − 2 cos(t)
dy/dt = 2 sin(t)
0 ≤ t ≤ 2π
The arc length formula is;
L = (α,β)∫√[(dx/dt)² + (dy/dt)²]dt
Where α and β are the boundary points. Thus, applying this to our question, we have;
L = (0,2π)∫√((1 - 2 cos(t))² + (2 sin(t))²)dt
L = (0,2π)∫√(1 - 4cos(t) + 4cos²(t) + 4sin²(t))dt
L = (0,2π)∫√(1 - 4cos(t) + 4(cos²(t) + sin²(t)))dt
From trigonometry, we know that;
cos²t + sin²t = 1.
Thus;
L = (0,2π)∫√(1 - 4cos(t) + 4)dt
L = (0,2π)∫√(5 - 4cos(t))dt
Using online integral calculator, we have;
L = 13.3649
