Final answer:
The surface integral of the function xyz over a cone is evaluated by computing the surface element ds from the given parametric equations, finding the normal vector, and then integrating over the specified ranges of u and v.
Step-by-step explanation:
To evaluate the surface integral of the function f(x,y,z) = xyz over the cone with the given parametric equations, we first need to compute the surface element ds. Given the parametric equations x = u cos(v), y = u sin(v), and z = u, where 0 ≤ u ≤ 3 and 0 ≤ v ≤ π/2, we can find the partial derivatives of x, y, and z with respect to u and v to obtain the tangent vectors to the surface. The cross product of these tangent vectors will give us the normal vector, which is then used to compute the magnitude of ds.
The surface integral of f over the cone is then the double integral of f(x(u,v),y(u,v),z(u,v)) times the magnitude of ds, integrated over the given ranges of u and v. This requires setting up and evaluating the integral by integrating u from 0 to 3 and v from 0 to π/2.
The final solution will be the result of this double integral, which is typically computed using standard calculus techniques such as substitution and partial integration.