Volume of solid with double integrals

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Homework Statement

find volume of solid bounded by z=x, y=x, x+y=2 and z=0

The Attempt at a Solution

first need to find domain.

for x bounds, when y=0, x=0, when y = x, x+x=2 so x=1 therefore 0 < x < 1
for y bounds, x < y < 2-x

now I am trying to work out what i integrate over. usually its the plane z, so if z=x then the integral is SS x.dxdy

is that right?

 

[ProPatto16]

now i know that's the right integral. but when i integrate first with respect to y, there is no y.

so just integrate x with respect to x which is x^2/2 then sub in 1 and the final answer is 1/2?

but I am not sure if its that simple?