Are you calculating the area bounded by the curves ? Surface area could imply the surface area of a volume (obtained by revolution), and would greatly change the answer.
Assuming you are just looking for bounded area, what do you mean by "centroid"? The area between the x-axis and the curve is just . That is the very definition of area under the curve.
Edit: Oh, by centroid, do you mean the "center" of that area? Let be a number such that and . Then the centroid is . You probably don't have to solve the resulting cubic, but if you did, you could show