Calculus assignment question
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
I calculated the y-component of the centroid wrong. I think it would actually be some value such that .
So, the actual centroid would be
There is NO "surface area" bounded by and the x-axis! I suspect you mean the surface area of the figure generated by rotating around the x-axis, with x between 0 and 2 (so y between 0 and 4).
We can write parametric equations for that surface by taking y and z to be and so that with r and as parameters, going from 0 to 4 and from 0 to . We can then write a point on that surface as a vector equation .
Now the derivatives, and are tangent to the surface. Their cross product, has length so that the "integral of surface area is .
The surface area is given by