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 = 1 and the planes z = x + 7 and z = 0.

This question is based on the divergence theorem.Therefore, the flux  is    by using  divergence theorem .

Given:

f(x, y, z) = , s is the surface of the solid bounded by the cylinder and the planes z = x + 7 and z = 0.

We need to determined the surface integral f · ds.

According to the question,

Let  D be the region whose boundary is . Then by the divergence theorem,

⇒

Convert the given coordinates into cylindrical coordinates.

We get ,

x = r cos

y = r sin

z = z

Then, the dV = r dr d dz.

Now the integral become,

Therefore, the flux  is    by using  divergence theorem .

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