outward flux by Divergence Theorem

Question

How I can calculate the outward flux of the vector field F(x,y,z)=2xi+yj+zk across the boundary of  z=x^2+y^2 paraboloid  and  z=2x the plane  by the Divergence Theorem ?

Answer 1

The divergence theorem says that the desired flux surface integral is equal to the volume integral of the divergence. of F. The divergence of F is a constant (equal to 4) .

Thus the problem boils down to the calculation of the volume between the plane and the paraboloid.

This volume can be computed using the slice method. The volume is thought of a a stack of slices each normal to the y axis. The thickness of each slice is dy. For any value of y, the area of each slice can be found as the area ( in the x,z plane) between the curves 2x and (x² + y²) . This involves an ordinary integral over dx of

[ 2x - (x² + y² ) ] with limits of 1 - sqrt(1 - y²) and 1 + sqrt(1 - y²) .

These limits are found by locating the intersection of the curves 2x and x² + y² .

After some simple calculus and a lot of algebra, the area (for a given value of y) is found to be

(4/3) ( 1 - y²)^3/2

The next step is the integration of this result over dy with limits of - 1 and + 1

I used a TI - 84 calculator to evaluate this to be (4/3) 1.178097228

The exact expression probably involves the arcsinh function.

The final answer is this result multiplied by 4.