Answer
a) $\oint F \cdot dr=0$ for any closed path.
b) $2 \pi$
Work Step by Step
(a) F is conservative, so $\oint F \cdot dr=0$ for any closed path.
(b) $\int_C \cdot dr=\int_{1,0,0}^{1,0,2 \pi} \nabla (x^2 ze^y) dr$
Now, $[(x^2 ze^y)]_{1,0,0}^{1,0,2 \pi}-[(x^2 ze^y)]_{1,0,0}^{1,0,2 \pi}=2 \pi -0=2 \pi$
