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The question explicitly asks you to use stokes theorem right ?

|dw:1438227332463:dw|

The normal vector would always just be <0, 0, 1>, right? Since it lies on the plane of z=2?

thats right, it doesn't change over the region

What do I substitute in for z in the integrand? Just 2?

the surface is entirely in plane z=2, so we can replace z by 2

np, i hope you have used the geometry to evaluate that double integral