curve integral - intersection between plane and sphere

I am going to calculate the line integral$$ \int_\gamma z^4dx+x^2dy+y^8dz,$$where $\gamma$ is the intersection of the plane $y+z=1$ with the sphere $x^2+y^2+z^2=1$, $x \geq 0$, with the orientation given by increasing $y$.

Since $\gamma$ is an intersection curve, I decided to use Stoke's theorem, applied to the vector field $(z^4,x^2,y^8)$ and an oriented surface $Y$ with boundary $\gamma$. But how am I going to parametrize the surface so I can use it with Stoke's thoerem?

If I parametrize the surface by $(x(s,t),y(s,t),z(s,t))=(0,t,1-t),$ $x^2+y^2+z^2\leq 1$, I will get the normal vector $(0,0,0)$, but the normal vector is going to point upwards, I think.