How to find double integral borders knowing the object's limitations?
Solution 1:
The projection of the region in XY plane is given by $x \geq 0, y \geq 0, x+y \leq 1$ (see the below diagram).
So the integral to find the volume is,
$ \displaystyle \int_0^1 \int_0^{1-y} (1 + x + y) ~ dx ~ dy = \frac 56$
OR
$ \displaystyle \int_0^1 \int_0^{1-x} (1 + x + y) ~ dy ~ dx$