Hi there, I just wandered in. I thought I had a fairly good idea of how to do #1, but I'm not too fond of the answer I keep finding. The problem in question is at the top of my scanned sheet dealing with work and line integral.
Note the vector field is conservative
$\displaystyle f(x,y,z)=\int2y^2zdx=2xy^2z+g(y,z)$
$\displaystyle f_y(x,y,z)=4xyz+g_y(y,z)=4xyz$
so g can only be a function of z
$\displaystyle f_z(x,y,z)=2xy^2+g_z(z)=2xy^2+1 \iff g_z(z)=1 \iff g(z)=z$
so our function is
$\displaystyle f(x,y,z)=2xy^2z+z$
by the fundemental theorem of line integrals we can just evaluate at the end points for the value.
$\displaystyle r(0)=(1,0,0)$
$\displaystyle r(\pi)=(-1,0,\frac{\pi}{6})$
$\displaystyle f(-1,0,\pi/6)-f(1,0,0)=2(-1)(0)^2(\pi/6)+\pi/6-(2(1)(0)^2(0)-0)=\frac{\pi}{6}$
I hope this helps.