I am given a differential form $$\omega = (y^2+z^2)dx + (z^2+x^2)dy + (x^2+y^2)dz$$ and I have to evaluate the integral $\int_\gamma \omega$ on the border of the set $$\gamma=\{(x,y,z)\in \mathbb{R^3} : x^2+y^2+z^2=2Rx ,\quad x^2+y^2 = 2rx, \quad z>0\}$$ going only once clockwise.
I think I am no able to find a parametrization of the border using only one variable, because parameterizing the cylinder I obtained $$\cases{x=r+r\cos\theta \\y=r\sin\theta\\z=z}$$ with $0\le\theta\le 2\pi$. Substituting this into the sphere I have $$z^2 = 2Rr(1+\cos\theta)-2R^2-2r^2\cos\theta$$ and given that z is positive $$z = \sqrt{2Rr(1+\cos\theta)-2R^2-2r^2\cos\theta}.$$ So I have $r(\theta) =(r+r\cos\theta, r\sin\theta, \sqrt{2Rr(1+\cos\theta)-2R^2-2r^2\cos\theta)}$ and this seems incorrect to me because then I am not able to proceed with the integral.
Where am I wrong?