The Jacobian for this transformation is

with determinant
, hence the area element becomes

Then the integral becomes

where
is the unit circle,

so that

Now you could evaluate the integral as-is, but it's really much easier to do if we convert to polar coordinates.

Then

Answer:
all done bud, I'm just here for the extra points :D
The unit rate is 2/3 because in the line y=2/3x, 2/3 is the slope (y=mx+b). The slope of the line is pretty much the definition of a unit rate because the unit rate is a constant addition or subtraction over and over. In addition, it is a straight line(y=mx+b) and the values are going up at a constant rate of 2/3.