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**Opalg** The intersection will be a curve, not a surface, and that is best described parametrically. Write the equation of the cylinder as $\displaystyle x^2 + (y-1)^2=1$, and you see that this can be parametrised as $\displaystyle x = \sin\theta,\:y = 1+\cos\theta$. Then $\displaystyle x^2 + y^2 = 2+2\cos\theta$, and if you substitute that into the equation of the sphere you see that $\displaystyle z^2 = 2(1-\cos\theta) = 4\sin^2(\theta/2)$.

Therefore the part of the curve in the positive octant can be described by the parametric representation $\displaystyle (x,y,z) = (\sin\theta, 1+\cos\theta, \sin(\theta/2))\;(0\leqslant\theta\leqslant\pi)$.