Question

The curve r(t)=[t sin(t)+cos(t), t cos(t)-sin(t), (sqrt(3)t^2)/2] can be reparametrized with respect to arc length s(t)=[ ] measured from the point P=(1,0,0)

Answer 1

To reparametrize the curve r(t) with respect to arc length s(t), we need to find the expression for s(t) by integrating the magnitude of the derivative of r(t) and integrate it from t = t1 to t = t2.

Step-by-step explanation:

To reparametrize the curve r(t) with respect to arc length s(t), we need to find the expression for s(t). The formula for arc length is given by As = ∫(t1 to t2) ||r'(t)|| dt, where r'(t) is the derivative of r(t) with respect to t. In this case, r(t) = [t sin(t)+cos(t), t cos(t)-sin(t), (sqrt(3)t^2)/2], so we need to find the derivative of r(t) and integrate it from t = t1 to t = t2 to get the expression for s(t).

First, let's find the derivative of r(t):

r'(t) = [d/dt(t sin(t)+cos(t)), d/dt(t cos(t)-sin(t)), d/dt((sqrt(3)t^2)/2)].

After finding the derivative, we can calculate the magnitude of r'(t) and integrate from t = t1 to t = t2 to get the expression for s(t).