Surely you know that the mass of an object is its density integrated over its volume.

Your volume is bounded by the cone , below by z= 0 and above by z= 1. The only thing peculiar about this is that the vertex of the cone is at z= 2. Since z<1 2- z is below 2 and we can use . Projecting onto the xy-plane, we get circle with center at (0,0) and radii 1 to 2.

In x, y, z coordinates, we would have to integrate over the 'washer' between the circle with radius 1 and the circle with radius 2. That can be done but would require four separate integrals (from x= -2 to x= -1, from x= -1 to 1 with y above the smaller circle, from x= -1 to 1 with y below the smaller circle, from x= 1 to 2). It is much easier to use cylindrical coordinates. In that case, r goes from 1 to 2, [itex]\theta[/itex] goes from 0 to 2\pi, and, for each r and [itex]\theta[/itex], z goes from the plane, z= 1 up to the cone, [tex]z= 2-\sqrt{x^2+ y^2}= 2- \sqrt{r^2}= 2- r.

In cylindrical coordinates the density function is .

That is your integral is .