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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  • kb
    Lv 7
    8 years ago
    Favorite 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!

  • 4 years ago

    Stokes Theorem Cylinder

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