Use Stokes' Theorem to evaluate S consists of the four sides of the pyramid with vertices (0, 0, 0), (3, 0, 0), (0, 0, 3), (3, 0,3) and (0, 3, 0) that lie to the right of the xz-plane, oriented in the direction of the positive y-axis.
A)1
B)16
C)49
D)0
E)12

Question

Use Stokes' Theorem to evaluate   S consists of the four sides of the pyramid with vertices (0, 0, 0), (3, 0, 0), (0, 0, 3), (3, 0,3) and (0, 3, 0) that lie to the right of the xz-plane, oriented in the direction of the positive y-axis.
A)1 
B)16 
C)49 
D)0 
E)12

Quizplus · Accepted Answer

We can use Stokes' Theorem to convert the surface integral over S to a line integral over its boundary curve. The boundary curve is the union of the four line segments that make up the edges of the pyramid. We can parameterize these line segments as: C1(t) = , 0 ≤ t ≤ 3 C2(t) = <3, 0, t>, 0 ≤ t ≤ 3 C3(t) = <3 - t, 0, 3>, 0 ≤ t ≤ 3 C4(t) = <0, t, 3 - t>, 0 ≤ t ≤ 3 Notice that the curves C1 and C3 have the same endpoints <0, 0, 0> and <3, 0, 3>, so they cancel out each other's contribution to the line integral. Similarly, the curves C2 and C4 have the same endpoints <3, 0, 3> and <0, 3, 0>, so they cancel out each other's contribution to the line integral. Therefore, we only need to consider the curves C1 and C4, which make up the boundary of the base of the pyramid. On C1: ds = <1, 0, 0> dt and F = <2t, 0, 0> ∫C1 F·ds = ∫0^3 <4t, 0, 0>·<1, 0, 0> dt = ∫0^3 4t dt = 6 On C4: ds = <0, 1, -1> dt and F = <0, 1, -1> ∫C4 F·ds = ∫0^3 <0, t, -t>·<0, 1, -1> dt = ∫0^3 2t dt = 3 Therefore, by Stokes' Theorem: ∫S curl(F)·dS = ∫C1 F·ds + ∫C2 F·ds + ∫C3 F·ds + ∫C4 F·ds = 6 + 3 - 6 - 3 = 0. So the best choice is D, 0.

Use Stokes' Theorem to Evaluate S Consists of the Four