From Rogawski ET section 173 exercise 15 Use the Divergence

From Rogawski ET section 17.3: exercise 15. Use the Divergence Theorem to evaluate the surface integral F. dS. F = 5x + y, z, 5z - x , S is the boundary of the region between the paraboloid Z = 36 - x2 - y2 and the xy-plane. F . dS =

Solution

S is given by (x^2) + (y^2) = 36

F.dS = (5x+y, z, 5z-x).(dx, dy, dz)

= (5x+y) dx + z dy + (5z-x) dz

Integral of F.dS over S is

x from -6 to 6, y from -6 to 6, z is 0.

= (5/2)(36-36) + Integral from -6 to 6 of (y dx)

Say, x = (6 sin a)

dx = (6 cos a) da

Integral from -6 to 6 of sqrt(36 - x^2) dx =

Integral from -pi/2 to +pi/2 of 36 (cos a)^2 da

= 36(0.5 pi) = 18 pi = 56.55