Question

∫γ(z√(1-x^2-y^2))ds. γ is the intersection circle between the sphere x^2+y^2+z^2=1 and plane x=y. can someone help me with the parametrization?

Answer 1

$$\large\int_\gamma z\sqrt{1-x^2-y^2}\,ds$$

Answer 2

the circle of intersection is trivial to find:$$x^2+y^2+z^2=1\\2x^2+z^2=1\\2y^2+z^2=1$$the circle equations permit trivial parameterization \(x=y=1/\sqrt2\ \cos t,z=\sin t\)