Answer
$\dfrac{-5}{6}$
Work Step by Step
As we are given that $F=yi+xzj+x^2k$
Applying Stoke's Theorem, we have
$\oint F \cdot dr=\iint _S (\nabla \times F) \cdot n d\sigma$
or, $\oint F \cdot dr=\int_0^{1} \int_0^{1-y} (-xi-2xj+z-k) (\dfrac{(i+j+k)}{\sqrt 3}) \sqrt 3 dA $
or, $\int_0^{1}[-2x^2+y^2 ]_0^{1-y} =\dfrac{-5}{6}$
