Finding Flux of Vector Field

Question

Find the flux of the vector field F→ (x, y, z) = <e^−y , −y, x sinz> across the positively oriented surface S defined by R^→ (u, v) = <2 cosv,sinv, u>, with 0 ≤ u ≤ 5, 0 ≤ v ≤ 2π

Answer 1

R_u = <0, 0, 1>; R_v = <- 2sinv, cosv, 0>

R_u× R_v = < - cosv, - 2sinv, 0>; F(u, v) = <e^-sinv, - sinv, 2cosvsinu>;Flux = ∫₀⁵∫₀^2πF(u,v)·(R_u× R_v)dudv =

∫₀⁵∫₀^2π(- e^-sinvcosv + 2sin²v)dudv = ∫₀⁵(e^-sinv + v - 05sin2v)₀^2πdu = ∫₀⁵[(1 + 2π - 0) - 1]u = 2π∫₀⁵du = 10π