Given
$\int_\gamma F \cdot dr$
where $F=x^2i+y^2j+z^2k$ where $\gamma$ is the intersection of the sphere $x^2+y^2+z^2=a^2$ with the plane $y=z$.
I know it sounds quite silly and easy to compute this but the exercise make the statement about the intersection its made in the first octant of the plane with the coordinates (0,0,a) to $(\frac{a}{\sqrt{2}}, \frac{a}{\sqrt{2}},0)$, so this confuses me. I used to make this with the eyes closed but its been time since i stop doing this even i know that this would be easier with Stokes Theorem, but can you please help me out, with a hint or any suggestions
You don't necessarily need Stokes' theorem. A more basic fact would be that $F$ is conservative
$$F = \nabla\left(\frac{x^3+y^3+z^3}{3}\right)$$
and the closed loop line integral of any conservative vector field will be $0$ by the fundamental theorem of line integrals.