Consider f: \mathbb R^2 \to \mathbb R with
f(x,y) = A x + B y.
Compute \displaystyle \int \int_{{\color{orange}D}} f(x,y) dA over the given
\color{orange}D below.
\displaystyle \int \int_{{\color{orange}D}} f(x,y) dA =
Parametrise the domain
{\orange{D}} \subset \mathbb R^2 with (for example)
{\orange{D}} = \left\{ (x,y) \, | \, -Y \leq y \leq Y, \;
0 \leq x \leq -fractionReduce(X,Y) |y| +
X\right\}.
With this one gets
\displaystyle \int\int_{\orange{D}} f(x,y) dA =
\int_{-Y}^{Y} \int_0^{f_O(y)} (Ax + By) \, dx dy where
f_O(y) = -fractionReduce(X,Y) |y| + X. (Cave: Order)
For the 1st (inner) integration (first w.r.t. x) we decompose the integral into y \geq 0
and y < 0:
\displaystyle \int_{-Y}^{Y} \int_0^{f_O(y)} (Ax + By) \, dx dy
=
\int_{-Y}^{0} \int_0^{fractionReduce(X,Y,small=true) y + X} (Ax + By) \, dx dy
+
\int_0^{Y} \int_0^{-fractionReduce(X,Y,small=true) y + X}
(Ax + By) \, dx dy.
Now apply \displaystyle \int (Ax + By) \, dx = fractionReduce(A,2)x^2 +
Byx +C.
This yields for the 2nd integration step:
\displaystyle
\int_{-Y}^{0} fractionReduce(B*X*Y+A*A*Y*Y,X*X) y^2 + B*X+2*A*Y y + A*X*X \, dy
+ \int_0^{Y} fractionReduce(-B*X*Y+A*A*Y*Y,X*X) y^2 + B*X-2*A*Y y + A*Y*Y \, dy
= fractionReduce(Y*Y*Y*(B*X*Y+A*A*Y*Y)*2 - 3*X*X*(B*X+2*A*Y)*(-Y)- 6*X*X*A*X*X*(-Y) ,6*X*X) + fractionReduce(Y*Y*Y*(-B*X*Y+A*A*Y*Y)*2 + 3*X*X*(B*X-2*A*Y)*Y+ 6*X*X*A*X*X*Y ,6*X*X) =
fractionReduce(Y*Y*Y*(B*X*Y+A*A*Y*Y)*2 - 3*X*X*(B*X+2*A*Y)*(-Y)- 6*X*X*A*X*X*(-Y) + Y*Y*Y*(-B*X*Y+A*A*Y*Y)*2 +
3*X*X*(B*X-2*A*Y)*Y+ 6*X*X*A*X*X*Y,6*X*X).