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Evaluate the surface integral S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = x i + y j + 7 k S is the boundary of the region enclosed by the cylinder x2 + z2 = 1 and the planes y = 0 and x + y = 6

Answer :

V is the 3D region enclosed by S. 

div(F)= ∂(x)/∂x + ∂(y)/∂y + ∂(10)/∂z = 2 

∫ = ∫∫∫ [V] div(F) dV = 2 ∫∫∫ [V] dV 

Using cylindrical coords, ( rcos(θ), y, rsin(θ) ), dV=rdrdθdy 

∫ = 2 ∫∫∫ rdrdθdy, [r=0,1], [θ=0,2π], [y=0,7−rcos(θ)] = 14π (easily integrated)
braza909

Answer:

z=(x^2+y^2)

given equation

x^2+z^2=1

and y=0 and

x+y=6

For a given problem for the given vector calculus,the final value of integration is 45.56

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