Volume bounded by $y=x$, $z=x$, $z=0$ and $x+y=2$

Question

I'm having trouble visualizing the domain I am trying to integrate over. May someone help me visualize so that I may set up the double integral. Thanks.

Answer 1

Take a look at this figure. The bold lines show were two planes intersect. Does this help?

Answer 2

With a slight rearrangement, you have a tetrahedron with boundary planes: $z=0, z=x$ and $y=x, y=2-x$ .

Thus you wish to integrate $z\in[0;x]$ and $y\in [2-x; 0]$ but where should $x$ lie?

$$\iiint\limits_{\{(x,y,z): x\in[\bbox[white, border:gray 1pt dotted]{\color{white}{~0~;~1~}}], y\in[2-x;x], z\in[0;x]\}}1\operatorname dz\operatorname d y\operatorname d x$$