Final answer:
The length of the curve can be found using the arc length formula. The integral of the magnitude of r'(t) over the interval [0, 4] will give the length.
Step-by-step explanation:
The length of the curve can be found using the arc length formula. In this case, the position vector r(t) is given as cos(t)i + sin(t)j + ln(cos(t))k. To find the length, we need to find the derivative of r(t) and evaluate it from t = 0 to t = 4. The derivative of r(t) is given by r'(t) = -sin(t)i + cos(t)j - tan(t)sec(t)k.
The length of the curve is then given by the integral of the magnitude of r'(t) over the interval [0, 4].To evaluate the integral, we can use a calculator or a computer program. The result will be the length of the curve.