Question

Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above.
F(x, y, z) = xyi + 5zj + 7yk, C is the curve of intersection of the plane x + z = 8 and the cylinder x2 + y2 = 9.

Answer 1

 ∫c F · dr 
= ∫∫s curl F · dS, by Stokes' Theorem 
= ∫∫ <2, 0, -x> · <-z_x, -z_y, 1> dA 
= ∫∫ <2, 0, -x> · <1, 0, 1> dA, since z = 2 - x 
= ∫∫ (2 - x) dA. 

Since the region of integration is inside x^2 + y^2 = 9, convert to polar coordinates: 
∫(r = 0 to 3) ∫(θ = 0 to 2π) (2 - r cos θ) * (r dθ dr) 
= ∫(r = 0 to 3) (2 * 2π - 0) * r dr 
= 2πr^2 {for r = 0 to 3} 
= 18π.