.$\int\int_{R}(x − 6y) dA$where R is the triangular region with vertices (0, 0), (5, 1), and (1, 5). x = 5u + v, y = u + 5v

Question

I would like to know why u is from 0 to 1 as highlighted in yellow from the following image.

Use the given transformation to evaluate the integral.$\int\int_{R}(x − 6y) dA$where R is the triangular region with vertices (0, 0), (5, 1), and (1, 5). x = 5u + v, y = u + 5v

Answer 1

$R(u,v)$ is the region bounded by $v=0$, $u=0$, and $u+v=1$, which are represented in the image below. Therefore: $$ R(u,v) =\{ (u,v)\;|\;0 \le v \le 1-u, 0 \le u \le 1 \} $$ Note that you could also write it as $$ R(u,v) =\{ (u,v)\;|\;0 \le u \le 1-v, 0 \le v \le 1 \} $$