Evaluate ʃʃ fds where
F(x,y,z)= xy/z and S={ (x,y,z) ∈ R3 : z=x^2+y^2 , z∈ [4,16] }

Question

Evaluate ʃʃ fds where 
F(x,y,z)= xy/z and S={ (x,y,z) ∈ R3 : z=x^2+y^2 , z∈ [4,16] }

anonymous · Answer

surface integral?

anonymous · Answer

yes it is surface integral. but how to do it??

anonymous · Answer

what is f?

anonymous · Answer

f(x,y,z)= xy/z

anonymous · Answer

The surface area S is a paraboloid, and for a 3-D objects with 2 symmetrical axes, cylindrical coordinate is preferable.

anonymous · Answer

Typo: The integrand should be: $$f(x,y,z)\sqrt {{{\left( {\frac{{\partial g}}{{\partial x}}} \right)}^2} + {{\left( {\frac{{\partial g}}{{\partial y}}} \right)}^2} + 1} $$

anonymous · Answer

Let z=g(x,y)=x^2+y^2 => f(x,y,z)=xy/(x^2+y^2)
Take partial derivatives of g wrt x,y resp.
The integration region is annular with inner radius = 2 and outer radius = 4