There is no itegration required here. Have you not noticed that skeeter gave the answer? Since the density is uniform, you are really looking for the "centroid". And the centroid of a triangle has coordinates that are the arithmetic average of coordinates of the vertices. Since the axis of symmetry is 4, one vertex is at (x, 4) for some x. The two other vertices are symmetrically placec around that line of symmetry so, since one is given as (2, 1), 3 units below the axis of symmetry, the other is at (2, 4+ 3)= (2, 7).
Of course, (4+ 1+ 7)/3= 4. All you need to do is solve (2+ 2+ x)/3= -3 for x to find the third point.
gbenguse78, the density "function" appears in the integrals for moment and area (the numerator and denominator). Since, in this problem, it is a constant, it can be taken out of each integral and will cancel. You can just ignore it.
Thanks. I am slightly worried about this paper. But we will get there.