Question

Part 1. A certain helix in three-dimensional space is described by the equations x = cos theta, y = sin theta, and z = theta. Calculate the line integral, along one loop of this helix, of the dot product V

Answer 1

Let's find the potential function f(x,y). Since ?f(x,y) = <f_x, f_y>, we can use this to help us.

If ?f = F(x,y), then

f_x = 2x^(3/5) [Equation 1]

f_y = e^(y/6) [Equation 2]

Integrate f_x [Equation 1] with respect to x, you get a proposed f(x,y) which is given by

f(x,y) = (5/4)x^(8/5) + g(y) [Equation 3]

We partially differentiate Equation 3 with respect to y and compare this (basically equate) to
Equation 2.

f_y(x,y) = g'(y) = e^(y/6).

So g(y) = integral of e^(y/6) dy = 6e^(y/6) + K (K = a constant).

Plug this g(y) into the Equation 3. Therefore,

f(x, y) = (5/4)x^(8/5) + 6e^(y/6) + K

This is our potential function. When we plug in the endpoints, we use K = 0. This is the thing with the Fundamental Theorem of Line Integrals, you just need the endpoints

So, the line integral is equal to f(0, 1) - f(1, 0) = [(0 + 6e^(1/6)) - ((5/4)(1) + 6(1)] [Do some simplifying]

= 6e^(1/6) - 29/4