Let S be the rectanglular prism with comers at (0,0,0), (0,1,0), (1,0,0), (1,1,0

Question

Let S be the rectanglular prism with comers at (0,0,0), (0,1,0), (1,0,0), (1,1,0), (0,0,2), (0,1, 2), (1,0,2), (1,1, 2). Sketch the surface Sin space with normals pointing out of the surface. Name the six faces of S:S_1, S_2, S_3, etc. Write out the six integrals needed to compute the flux through S.DON'T EVALUATE THE INTEGRALS. The Divergence Theorem relates a surface integral to a triple integral. Write out the Divergence Theorem (in general. not just for this problem) Apply the Divergence Theorem to this problem to calculate the flux out of volume bounded by S.

Explanation / Answer

We compute ZZ R 1 + (x 1)2 + 4y 2 dA, where R = [0, 3] × [0, 2], in two ways. First, Z 3 0 Z 2 0 1 + (x 1)2 + 4y 2 dy dx = Z 3 0 y + (x 1)2 y + 4 3 y 3 2 0 dx = Z 3 0 2 + 2(x 1)2 + 32 3 dx = 2x + 2 3 (x 1)3 + 32 3 x 3 0 = 6 + 2 3 · 8 + 32 3 · 3 (0 1 · 2 3 + 0) = 44. In the other order: Z 2 0 Z 3 0 1 + (x 1)2 + 4y 2 dx dy = Z 2 0 x + (x 1)3 3 + 4y 2x 3 0 dy = Z 2 0 3 + 8 3 + 12y 2 + 1 3 dy = 3y + 8 3 y + 4y 3 + 1 3 y 2 0 = 6 + 16 3 + 32 + 2 3 = 44.

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