Let be $$f(x,y)=24y(1-x-y) \ \text{ if } x,y\geq 0, x+y<1; 0 \text{ elsewere}$$
Find de CDF, $F(x,y)$.
Solution:
$F(x,y)=0$ for $(x,y)$ in quadrants II, III and IV.
Let be $R$ the region suh that $x,y\geq 0, x+y<1$.
If $(x,y)\in R$, $$F(x,y)=\int_0^x\int_0^y24v(1-u-v)dvdu=12xy^2-6x^2y^2-8xy^3. $$
what would be the integration limits for points outside $ R $?
The interval you require is:
$$\{{\langle u,v\rangle:} {{0\leq u\leq \min[x,1],} {0\leq v\leq \min[y, 1-u]}}\}$$