Question

Consider the curve r=(etcos(−6t),etsin(−6t),et). Compute the arclength function s(t)

A) s(t) = ∫ eᵗ dt
B) s(t) = ∫ √(e(²ᵗ) + 36) dt
C) s(t) = ∫ e(ᵗ/²) dt
D) s(t) = ∫ 6eᵗ dt

Answer 1

To compute the arclength function, use the formula s(t) = ∫√(x'(t)² + y'(t)² + z'(t)²) dt, where x'(t), y'(t), and z'(t) are the derivatives of x(t), y(t), and z(t) respectively. In this case, s(t) = ∫√(e^(2t) + 36) dt.

Step-by-step explanation:

To compute the arc length function, we will use the formula
s(t) = ∫√(x'(t)² + y'(t)² + z'(t)²) dt
where x'(t), y'(t), and z'(t) are the derivatives of x(t), y(t), and z(t) respectively. In this case, the derivatives are
x'(t) = e^(t)cos(-6t) + e^(t)(-6)sin(-6t)
y'(t) = e^(t)sin(-6t) + e^(t)(-6)cos(-6t)
z'(t) = e^(t)
Substituting these derivatives into the arc length formula and simplifying, we obtain the correct option:
s(t) = ∫√(e^(2t) + 36) dt