work done by a vector field from a potential function

Question

I'm trying to find the work done using a scalar potential however the work done from a potential function never matches a line integral. do I not understand something, why cant I get this right?

$$ F = (\sin(x), 0, 2z) \\ r = (\rho, \rho, \rho^2) \hspace{1cm} 0 \leq \rho \leq \pi $$ Find the work by constructing a scalar potential.

These are my attempts:

Answer 1

Your second attempt was (aside from some notational mistakes) fine until the very end, when you computed $f(0,0,2\pi)$ instead of $f(\pi,\pi,\pi^2)$.

Answer 2

When you compute the integral without the potential, note that $$ -\cos(\pi) - \cos(0) = 0. $$ So, your answer should be $\pi^4$. Then, using the potential $f(x,y,z) := -\cos(x) + z^2$, you have $$ f(\pi, \pi, \pi^2) - f(0,0,0) = \pi^4. $$