Final answer:
To evaluate the line integral on the given circular helix, we calculate the differential arc length and use it to compute the integral. Over one full turn of the helix, the integral of the provided function is zero because the integral of sin(t) over 0 to 2\pi is zero.
Step-by-step explanation:
The question asks us to evaluate the line integral of a function along a given path, which in this case is a circular helix. The given helix is represented parametrically by (x = cos(t)), (y = sin(t)), (z = t). We are asked to integrate the function y sin(z) along this curve denoted by C. To do this, we first need to understand what ds, or the differential arc length element, represents on this path. It's the infinitesimal distance along the helix, which we need to calculate before integrating.
For a parametric curve described by (x(t), y(t), z(t)), the arc length differential ds is given by ds = \sqrt{(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2} dt. Substituting x(t) = cos(t), y(t) = sin(t), z(t) = t, we can compute the derivatives and find ds. Going through the integral computation, we realize that over one full turn of the helix, the integral of sin(t) from 0 to 2\pi is zero due to the periodic nature of the sine function, implying that the integral's value will also be zero. Therefore, the correct answer to this question is a) (0).