Example 3

EXAMPLE 3

Evaluate the surface integral

where S is the surface z = e^x-y over the region D given by

0 ≤ x ≤ 1, x ≤ y ≤ 1,

S is oriented with the top side positive, and

F(x, y, z) = 2i + j + z²k.

The region is sketched in Figure 13.5.8. The first step is to find ∂z/∂x and ∂z/∂y.

Figure 13.5.8

By definition of surface integral,

The same surface integral with S oriented with the top side negative has minus the above value.