Answer:
8√2
2√2
Explanation:
Using arc length for parametric equations:
s = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt
x = 2 sin²t
dx/dt = 4 sin t cos t
y = 2 cos²t
dy/dt = -4 sin t cos t
s = ∫ √((4 sin t cos t)² + (-4 sin t cos t)²) dt
s = ∫ √(32 sin²t cos²t) dt
s = ∫ 4√2 sin t cos t dt
s = ∫ 2√2 sin(2t) dt
s = -√2 cos(2t)
If we evaluate this between t=0 and t=2π, we get s = 0. This means the particle travels forward and backward on the same curve.
sin²t and cos²t have periods of π, so if we evaluate from t=0 to t=π/2:
s = -√2 cos(2 (π/2)) − -√2 cos(0)
s = √2 + √2
s = 2√2
So the particle travels 2√2 four times from t=0 to t=2π.
The distance traveled is 8√2. The length of the curve is 2√2.
We can check our answer by eliminating the parameter.
x + y = 2 sin²t + 2 cos²t
x + y = 2
y = 2 − x
The curve is the line y = 2 − x from (0, 2) to (2, 0). Using distance formula to find the length of the line:
d = √((x₂ − x₁)² + (y₂ − y₁)²)
d = √((2 − 0)² + (0 − 2)²)
d = 2√2