That's going to be a difficult problem but I think I would be inclined to rotate the coordinate system so as to make the line y= x correspond to the new y-axis. That is, we must rotate so that (x, y)= (0, 1) becomes (because that lies on the line y= x and has distance 1 from the origin) and (x,y)= (1, 0) becomes (because that lies on y= -x, the line perpedicular to y= x, in the fourth quadrand and has distance 1 from the origin).
Since a rotation is linear, we must have x= ax'+ by' and y= cx'+ dy'. Taking x= 0, y= 1, , , those become (i) and (ii) . Taking x= 1, y= 0, , , those become (iii) and (iv) .
Adding (i) and (iii) gives so . (i) is the same as b= -a so .
Adding (ii) and (iv) gives so . (iv) is the same as d= c so .
That is, and .
Now, the region is bounded by and y= x itself. Obviously, y= x becomes just y'= 0. becomes .
I will leave it to you to multiply that out. You are now rotating that around the x' axis from x'= 0 to .