Question
Asked Dec 9, 2019
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Use the Divergence Theorem to calculate the surface integral 

 
 
S
F · dS;

 that is, calculate the flux of F across S.

F(x, y, z) = x4i − x3z2j + 4xy2zk,

S is the surface of the solid bounded by the cylinder 
x2 + y2 = 4
 and the planes 
z = x + 3 and z = 0.
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Expert Answer

Step 1

Apply the Divergence Theorem as follows.

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F.dS = [[[ divFdV divF =V (x*i-x°z*j+ 4xy°zk) -x²=² + ôy - 4.xy²z = 4x + 4xy? = [[[(4x* + 4xy° )c&rdycz → > ||| divFdV

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Step 2

Use the cylindrical coordinates as ...

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J|F-as = [[f divFdV x =rcos0, y =rsin0 and - =:. →:=x+3 becomes :=rcos e +3 →0<:<rcos 0 +3, 0<r<2 and 0<0<2n = [|| 4x(x² +y² )dxdydz → [[[ divFdV 2x 2 r cosô+3 = [| | 4rcos e(r² )dzrdrde 2x 2 r cose+3 = || | 16r* cos Oczdrde 2x 2 r cos 8+3 16r*cose[:* drd® = [ = [[16r*cose[rcose +3]drde %3D

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