I am trying to compute the volume of the figure limited by these surfaces: $$z=xy$$ $$x+y+z=1$$ $$z=0$$
My attempt:
Using the first two equations I get $y=\frac{1-x}{1+x}$, which is, I think, the projection in the $XY$ plane of the intersection between these two surfaces (am I right?). Taking this into account I clonclude that the volume ($V$) is: $$V=\int_0^1\int_0^{\frac{1-x}{1+x}}xydydx+\int_0^1\int_{\frac{1-x}{1+x}}^{1-x}(1-x-y)dydx$$
My question:
I have a few doubts about this exercise. Firstly, I would like to know if what I have done is correct (I am concerned that the part about the projection may be wrong).
Secondly, is there another way to solve this exercise in which the integrals are a bit easier? (I think I know how to solve the ones I get, but it takes me some time to do them by hand), maybe I could have used triple integrals?
And finally, my main problem with this kind of exercise is that I do not manage to imagine the figure I am working with (which is what I do when I compute areas of shapes in two dimensions). When it is just spheres, planes and cylinders, I more or less manage to approach the problem successfully. However, when surfaces like $z=xy$ appear I start to have problems choosing the limits of integration. In this particular problem I could only come up with my solution (I am not even sure it is correct) because I used Geogebra to see what the figure looked like. Could someone suggest a way to approach this kind of problem? Is there a way to solve it without having to know what the figure looks like? Or should I learn more about quadrics in order to be able to picture them in my head?
Any help will be very useful.
Thanks in advance.