Final answer:
To integrate the function G(x, y, z) = xyz over a triangular surface, use a double integral over a parameterized region in the xy-plane.
Step-by-step explanation:
To integrate the function G(x, y, z) = xyz over a triangular surface with vertices (1,0,0), (0,2,0), and (0,1,1), you can use the concept of surface integrals. The surface integral over the triangular surface can be expressed as a double integral over a parameterized region in the xy-plane.
First, parameterize the triangular surface using two variables u and v that vary on the xy-plane. Then, express x, y, and z in terms of u and v using the given vertices.
Next, calculate the cross product of the partial derivatives of x and y with respect to u and v to obtain the normal vector of the surface. Finally, evaluate the double integral over the parameterized region using the function G(x, y, z) = xyz and the calculated normal vector.
The answer will give you the value of the integral over the triangular surface.