( D ) is the portion of the solid cone ( z=sqrt{x^{2}+y^{2}} ) between the planes ( z=2 ) and ( z=4 ). Use the Divergence Theorem to find the flux of the vector field ( mathbf{F}(x, y, z)=leftlangle 4 y^{2} z, frac{7}{48} y^{3}, frac{7}{16} x^{2} zrightrangle ) over the surface of ( D ). Consider cylindrical coordinates. Write the exact answer. Do not round.

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Step:1 ：(nabla cdot mathbf{F} = 2y^{2} + frac{21}{16}x^{2} )Step:2 ：( Phi(mathbf{F}) = int_{0}^{2pi} int_{2}^{4} int_{0}^{r} left(2y^{2} + frac{21}{16}x^{2}right)rdzdtheta drdy = int_{0}^{2pi} int_{2}^{4} int_{0}^{r} left(2r^{2} + frac{21}{16}r^{2}right)rdzdr )Step:3 ：(Phi(mathbf{F}) = int_{0}^{2pi} int_{2}^{4} left(frac{35}{8}r^{4}right) dr dtheta = frac{140}{3}(pi - 0) = frac{140pi}{3})

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Step: 1 ：(nabla cdot mathbf{F} = 2y^{2} + frac{21}{16}x^{2} )Step: 2 ：( Phi(mathbf{F}) = int_{0}^{2pi} int_{2}^{4} int_{0}^{r} left(2y^{2} + frac{21}{16}x^{2}right)rdzdtheta drdy = int_{0}^{2pi} int_{2}^{4} int_{0}^{r} left(2r^{2} + frac{21}{16}r^{2}right)rdzdr )Step: 3 ：(Phi(mathbf{F}) = int_{0}^{2pi} int_{2}^{4} left(frac{35}{8}r^{4}right) dr dtheta = frac{140}{3}(pi - 0) = frac{140pi}{3})

D is the portion of the solid cone z=x2+y2 between the planes z=2 and z=4. Use the Divergence Theorem to find the flux of the vector field F(x,y,z)=⟨4y2z,748y3,716x2z⟩ over the surface of D. Consider cylindrical coordinates. Write the exact answer. Do not round.

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