Use Stokes\' Theorem to evaluate integral_C (4xy i - 2zj + jk) middot dr where C
ID: 2878662 • Letter: U
Question
Use Stokes' Theorem to evaluate integral_C (4xy i - 2zj + jk) middot dr where C is the intersection of the plane x + z = 8 and the cylinder x^2 + y^2 = 4 oriented counterclockwise as viewed from above. Since the ellipse is oriented counterclockwise as viewed from above the surface we attach is oriented upwards. curl (4x y i - 2zj + yk) = The easiest surface to attach to this curve is the interior of the cylinder that lies on the plane x + z = 8. Using this surface in Stokes' Theorem evaluate the following. integral_C F middot dr = integral^x_2_x_1 integral^y_2_y_1 dy dx where y_1 = y_2 = x_1 = x_2 = Evaluate integral_C F middot dr =Explanation / Answer
Curl of f =<3,0,-4x>
c F · dr
= s curl F · dS, by Stokes' Theorem
= <3, 0, 4-x> · <-z_x, -z_y, 1> dA
= <3, 0, -4x> · <1, 0, 1> dA, since z = 8 - x
= (3 - 4x) dA
Now, convert to polar coordinates (due to x^2 + y^2 = 4):
= (r = 0 to 2) ( = 0 to 2) (3 - 4r cos ) * (r d dr)
= (r = 0 to 2) (3 * 2 - 0) * r dr
= 6r^2/2 {for r = 0 to 2}
= 12.
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