Weird ways of doing closed loop integrals

[wahaj]

I was looking at an example where it was evaluating a closed loop integral of a vector field around a triangle (0,0) (1,1) (2,0) by using greens theorem. This example was in the green's theorem section of the book so green's theorem must be used. Anyways the double integral was set up as follows
\int \int_T (y+x) dA
This integral wasn't evaluated the way I have been taught before but rather the book used the following formula
(\bar{y}+\bar{x}) \times area \ of \ triangle
In another instance a closed loop integrals of a vector field around F was evaluated and the formula used there was
(area of circle) ∇F \bullet \widehat{n}
I have never seen these formulas before and the book offers no explanation. Where does this formula come from?
On a side note does green's theorem evaluate the closed loop integral in the counterclockwise direction?

 

[SteamKing]

By applying Green's Theorem in the plane, you can demonstrate the first formula. After all, it is just a mathematical statement of the first moment of area of the region T about the x-axis added to the first moment of area of the region T about the y-axis. By definition, the first moment of any plane figure is the area of the figure multiplied by its centroidal distance.

I suspect a similar application of Green's Theorem for the second example will produce the formula shown.

 

[wahaj]

Looking back at the centroid formulas from last year I was able to figure out the first formula. Thanks for help.