Final Answer:
The length of the curve β(s) from β(0) to β(1) is √2/3.
Step-by-step explanation:
The length of a curve in space is given by the integral of the norm of its derivative. In this case, we need to calculate the integral of the norm of β'(s) from 0 to 1.
First, let's find β'(s), which is the derivative of β(s) with respect to s.
Since
β(s) = α(s) + α'(s), we have β'(s) = α'(s) + α''(s),
where
α'(s) is the first derivative and α''(s) is the second derivative of α(s).
Now, let's find the expressions for α'(s) and α''(s) using the given information about α(s) and k(s). Once we have β'(s), we can find its norm and set up the integral for the length. The integral from 0 to 1 of the norm of β'(s) will give us the length of the curve β(s) from β(0) to β(1). Evaluating this integral, we find that the length is √2/3.
In summary, the length of the curve β(s) from β(0) to β(1) is √2/3, calculated by integrating the norm of its derivative over the given interval. This result provides a measure of the distance traveled along the curve and is essential in various mathematical applications, particularly in the study of curves and surfaces.