$C$ be the curve of intersection of cylinder $x^2+y^2=2y$ and plane $y=z$ ; to evaluate $\int_C (y+z)dx + (x+z) dy +(x+y) dz$ by Stoke's theorem ?

Question

Let $C$ be the curve of intersection of the cylinder $x^2+y^2=2y$ and the plane $y=z$ ; how to evaluate $\int_C (y+z)dx + (x+z) dy +(x+y) dz$ by Stoke's theorem ? I cannot determine what is my surface $S$ over which to apply Stoke's theorem . Please help , thanks in advance

Answer 1

Hint: $x^2+y^2 = 2y$ can be rewritten into $x^2 +(y-1)^2 = 1$ and the parametric for of this would be the below.

$$\bar r(t) = cost\hat i + (sint+1) \hat j + (sint+1) \hat k$$ $$\bar r'(t) = -sint\hat i + cost \hat j + cost \hat k$$

$$F(\bar r(t)).\bar r'(t) = [2(sint+1)(-sint) + (cost+sint+1)cost + (cost + sint +1)cost] $$

$$\int\int curlF.dS = \int_{0}^{2\pi}F(\bar r(t)).\bar r'(t)dt $$

Which when you evaluate gives you $\boxed{0}$