Calculate ∬S f(x,y,z) dS for part of the surface x=z^3, where 0 ≤ x, y ≤ 7; f(x,y,z) = x.
Added by Victor Manuel B.
Step 1
Take r(x,y) = ⟨x, y, x^{1/3}⟩, 0 ≤ x ≤ 7, 0 ≤ y ≤ 7. Then r_x = ⟨1, 0, (1/3)x^{-2/3}⟩, r_y = ⟨0, 1, 0⟩, and r_x × r_y = ⟨- (1/3)x^{-2/3}, 0, 1⟩, so ||r_x × r_y|| = sqrt(1 + (1/9)x^{-4/3}). Show more…
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