Verify that the Divergence Theorem is true for the vector field F on the region E.?

∫∫ F⋅ dS = ∫∫∫ div F dV

div F = x

∫∫∫ x dx dy dz

convert to cylindrical

y = r cos t

z = r sin t

x = x

dx dy dz = r dx dr dt

∫[0,2pi] ∫[0,3]∫[0,2] xr dx dr dt

1/2 (2^2)(1/2)(3^2)(2pi)

18 pi

∫∫ F⋅ dS

this has to be done in three pieces. The wall of the cylinder, and the top and bottom.

The bottom

x = 0

∫∫ <0,-y,z><-1,0,0> dy dz

∫∫ 0 dy dz

0

top cap.

∫∫ <2,-y,z><1,0,0> dy dz

∫∫ 2 dy dz

18 pi

about the wall

y = 3 cos t

z = 3 sin t

x = x

dS = <dx/dt,dy/dt,dz/dt>x<dx/dx,dy/dx,dz/dx>

<0, -3 sin t, 3 cos t>X<1,0,0>

<0,3sin t, 3 cos t>

∫ <x^2, -3 cos t, 3 sin t>⋅<0,3sin t, 3 cos t> dx dt

∫ 0 dx dt

0

18pi = 18 pi