Final answer:
The length of the curve for the given vector-valued function can be found by integrating the magnitude of the derivative of the function over the desired time interval. The derivative yields the velocity vector, the magnitude of which gives us the speed function that needs to be integrated to find the arc length.
Step-by-step explanation:
The length of the curve defined by the vector-valued function r(t) = 8t, t², 1/12 t³ can be found by integrating the magnitude of the function's derivative from the lower to the upper limit of the interval for which we want to find the length. First, we calculate the derivative dr/dt, which gives us the velocity vector. Then we find the magnitude of this velocity vector, which gives us the speed as a function of time. By integrating this speed function over the desired interval, we obtain the arc length of the curve.
To start, we differentiate each component of r(t) with respect to time (t) to find the velocity vector v(t). We then determine the speed function by taking the square root of the sum of the squares of the components of v(t). Integrating this speed function over the desired time interval will yield the length of the curve.
The process can be mathematically described as follows:Compute dr/dt = (d/dt [8t], d/dt [t²], d/dt [1/12 t³]).Calculate the speed function as the magnitude of dr/dt, which is √((d/dt [8t])² + (d/dt [t²])² + (d/dt [1/12 t³])²).Integrate the speed function over the interval [a, b] to find the curve length.Please note that specific limits of integration were not provided. Hence, a generic method is explained without actual integration.