The distance traveled along the unit circle from t=0 to t=π equals which value?

The distance traveled along the unit circle from t=0 to t=π equals which value?

• A. π
• B. π/2
• C. π/4
• D. 1

Answer: (A)

On the unit circle, arc length corresponds to the angle traveled in radians, because the speed along the standard parameterization x = cos t, y = sin t is 1. The distance from t = 0 to t = π is the integral of the speed: ∫ from 0 to π of |r′(t)| dt, where r′(t) = (-sin t, cos t). The magnitude is √(sin^2 t + cos^2 t) = 1, so the distance is ∫_0^π 1 dt = π. Equivalently, this is half the circumference of the unit circle, since the full circumference is 2π. So the distance traveled is π.

On the unit circle, arc length corresponds to the angle traveled in radians, because the speed along the standard parameterization x = cos t, y = sin t is 1. The distance from t = 0 to t = π is the integral of the speed: ∫ from 0 to π of |r′(t)| dt, where r′(t) = (-sin t, cos t). The magnitude is √(sin^2 t + cos^2 t) = 1, so the distance is ∫_0^π 1 dt = π. Equivalently, this is half the circumference of the unit circle, since the full circumference is 2π. So the distance traveled is π.