Final answer:
The length of the curve r(t) = 4ti + t^3j + tk from t = 0 to t = 1 can be found by integrating the magnitude of the derivative of r(t) with respect to t over the given interval.
Step-by-step explanation:
The length of a curve can be found using the arc length formula. In this case, the curve is defined by the vector function r(t) = 4t^2i + t^3j + tk. To find the length of the curve from t = 0 to t = 1, we need to integrate the magnitude of the derivative of r(t) with respect to t over the given interval.
The derivative of r(t) is dr(t)/dt = 8ti + 3t^2j + k. The magnitude of dr(t)/dt is |dr(t)/dt| = sqrt(8t^2 + 9t^4 + 1). To find the length of the curve, we integrate |dr(t)/dt| with respect to t over the interval [0, 1].
The length of the curve is given by L = ∫ |dr(t)/dt| dt = ∫ sqrt(8t^2 + 9t^4 + 1) dt. This integral can be difficult to evaluate, but it can be approximated using numerical methods if an exact answer is not required.