Let

f(x, y, z) = z tan? 1(y2)i + z3 ln(x2 + 7)j + zk.

find the flux of f across s, the part of the paraboloid

x2 + y2 + z = 27

that lies above the plane

z = 2

and is oriented upward.

\int \int_{s}^{} f*ds

has divergence

If includes the disk which is contained in the paraboloid, then by the divergence theorem the flux is

If the disk is not part of , then subtract the flux of across the disk (with downward orientation; call it ). Parameterize by

where and , with normal vector

Then the flux of across is

which makes the flux across , .

5x+40

5(x + 8)

7x - 49

7(x - 7)

18x + 9

9(2x + 1)

15x - 10

5(3x - 2)

step-by-step explanation:

for number one all you have to do is factor 5 out of 5x + 40

for number 2 factor 7 out of 7x - 49

for number three factor 9 out of 18x + 9

for number four factor 5 out of 15x - 10