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# Use Stokes' Theorem to evaluate integral F · dr where C is oriented counterclockwise as viewed from above.?

F(x, y, z) = xyi + 3zj + 5yk,

C is the curve of intersection of the plane

x + z = 2

and the cylinder

x2 + y2 = 9.

Help would be much appreciated, thanks

### 2 Answers

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- kbLv 78 years agoFavorite Answer
∫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.

I hope this helps!

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