Final answer:
To find the length of the curve represented by r(t) = 2ti + etj + e^(-t)k, we must compute r'(t), find the magnitude of r'(t), and integrate this magnitude from 0 to 6.
Step-by-step explanation:
To find the length of the curve defined by the vector function r(t) = 2ti + etj + e(-t)k, we need to calculate its integral over the given interval from 0 to 6. This involves finding the derivative r'(t) and then integrating the magnitude of r'(t) over the specified interval:
Compute the derivative r'(t) of the vector function.Find the magnitude of r'(t) which gives the speed function.Integrate the speed function from t = 0 to t = 6 to get the arc length.Let's go through these steps:The derivative r'(t) is ((2)i + (et)j + (-e(-t))k).The magnitude of r'(t) is √(4 + e2t + e(-2t)).Finally, the integral of the speed function from 0 to 6 gives us the length of the curve.The steps are clear, but without performing the actual computations, we cannot give a numerical answer here.