Given the triangle \color{orange}D.
\displaystyle \int \int_{{\color{orange}D}} f(x,y) \, dA = \ldots
\displaystyle
\int_{-Y}^{0} \int_0^{fractionReduce(X,Y,small=true) y + X} f(x,y) \, dx dy
+
\int_0^{Y} \int_0^{-fractionReduce(X,Y,small=true) y + X} f(x,y) \, dx dy
\displaystyle
\int_{-Y}^{0} \int_0^{fractionReduce(X,Y,small=true) y + X} f(x,y) \, dy dx
+
\int_0^{Y} \int_0^{-fractionReduce(X,Y,small=true) y + X} f(x,y) \, dy dx\displaystyle
\int_{-X}^{0} \int_0^{fractionReduce(X,Y,small=true) y + X} f(x,y) \, dy dx
+
\int_0^{X} \int_0^{-fractionReduce(X,Y,small=true) y + X} f(x,y) \, dy dx\displaystyle
2 \int_0^{X} \int_0^{-fractionReduce(X,Y,small=true) y + X} f(x,y) \, dx dy\displaystyle
\int_{-Y}^{Y}
\int_{fractionReduce(X,Y,small=true) y - X}^{-fractionReduce(X,Y,small=true) y + X} f(x,y) \, dx dyA paramatrisation of
{\orange{D}} \subset \mathbb R^2 could be given by
{\orange{D}} = \left\{ (x,y) \, | \, -Y \leq y \leq Y, \;
0 \leq x \leq -fractionReduce(X,Y) |y| +
X\right\}.
Using and applying the defintion of the integral yield
\displaystyle \int\int_{\orange{D}} f(x,y) dA =
\int_{-Y}^{Y} \int_0^{f_O(y)} f(x,y) \, dx dy with
f_O(y) = -fractionReduce(X,Y) |y| + X. (Cave: Order)
For the 1st (inner) integration (w.r.t. x)
we decompose into y \geq 0 and y < 0:
\displaystyle \int_{-Y}^{Y} \int_0^{f_O(y)} f(x,y) \, dx dy
=
\int_{-Y}^{0} \int_0^{fractionReduce(X,Y,small=true) y + X} f(x,y) \, dx dy
+
\int_0^{Y} \int_0^{-fractionReduce(X,Y,small=true) y + X} f(x,y) \, dx dy.