Final answer:
To find the length of the curve, first find the derivative of the vector-valued function r(t). Then, calculate the magnitude of the derivative. Finally, integrate the magnitude over the given interval to find the length of the curve.
Step-by-step explanation:
The length of the curve can be found using the arc length formula. The formula for arc length is given by:
arc length = ∫ ||r'(t)|| dt
Where r(t) is the vector-valued function that represents the curve and r'(t) is its derivative. In this case, r(t) = cos(2t)i + sin(2t)j + 2ln(cos(t))k. Therefore, the first step is to find the derivative of r(t) with respect to t.
Once the derivative is obtained, the next step is to find its magnitude:
||r'(t)|| = sqrt((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2)
Now we can integrate the magnitude of r'(t) over the given interval to find the length of the curve.