Use Stokes' Theorem to evaluate
integral(c) F * dr
C is oriented counterclockwise as viewed from above.
F(x,y,z)= xyi + 2zj + 3yk
C is the curve of intersection of the plane x + z = 5 and the cylinder x^2 + y^2 = 9
integral integral [-(dR/dy - dQ/dz)dz/dx - (dP/dz - dR/dx)dz/dy + (dQ/dx - dP/dy)] dA
When i used this i got:
[-(3-2)-(0)+(-x)] = -1-x
I tried this another way and got:
(3y d/dy - 2z d/dz) - (3y d/dx - xy d/dz) + (2z d/dx - xy d/dy)
= (3 - 2) - (0) + (0-x) = 1 - x
So, i'm not sure whether the correct answer for that part is
(-1-x) or (1-x)
I'm thinking that (1-x) is correct because it is oriented counterclockwise.
So, if that is the case i get:
integral integral (1-x)dA
= integral(0, 2pi) integral(0, 3) (1-rcos(theta))r dr d(theta)
Did i do this correctly?