proof by stokes theorem

Question

use stokes theorem to evaluate F*dR where F(x,y,z)= xyi+x^2j+z^2k and C is the intersection of paraboloid z=x^2+y^2 with plane z=y with counterclockwise orientation looking down the positive z axis.

Answer 1

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Answer 2

By Stokes theorem,

∫_c F·dr = ∫_s∫curl F · N dS

curl F = 0i + 0j +  xk

N = -z_xi + -z_yj + k = -2xi - 2yj + k

So,

∫_s∫curl F · N dS

= ∫_s∫ x dx dy, where S is the area formed by x² + y² = y ==> x² + (y-0.5)² = 0.5²

Can you finish it?