Final answer:
To find the total arc length, calculate the integral of the magnitude of the derivative of r(t) for the interval 0 to π/4. The derivative r'(t) is calculated first, followed by its magnitude, and then the integral is evaluated to obtain the arc length.
Step-by-step explanation:
The question involves finding the total arc length of a parametric curve represented by r(t) = ⟨6cos(t), 6sin(t), 6ln(cos(t))⟩ for 0 ≤ t ≤ π/4. The arc length of a curve in space for a parametric equation r(t) from t = a to t = b is given by the integral of the magnitude of the derivative of r(t) with respect to t, or ∫||r'(t)||dt from t = a to t = b. First, calculate the derivative of r(t) which yields r'(t) = ⟩-6sin(t), 6cos(t), -6tan(t)⟩. Then, find the magnitude ||r'(t)||, which simplifies to √(√(36cos^2(t) + 36sin^2(t) + 36tan^2(t))). Substituting and evaluating the integral will give the total arc length for the specified interval.