F(x, y, z) = z tan−1(y2)i + z3 ln(x2 + 3)j + zk. find the flux of f across s, the part of the paraboloid x2 + y2 + z = 18 that lies above the plane z = 2 and is oriented upward. s f · ds =

By the divergence theorem, the flux of across the *closed* surface combined with the plane is given by a volume integral over the closed region:



So in fact, to find the flux over alone, we'll need to subtract the flux of over the planar portion, oriented outward. First, compute the volume integral by converting to cylindrical coordinates:






If the surface does actually contain , then you can stop here; otherwise, continue.

Now, parameterize the part of the *closed* surface in by



where and . We get a surface element



We don't need to worry about the first two components of

and so the surface integral over this region is



Then the total flux over alone is .