What Is the Centroid of the Region Bounded by y = (4x-x^2) and y = x?

[cwbullivant]

Homework Statement

Find the centroid of the region bounded by the curves y = (4x-x^2) and y = x.

Homework Equations

Centroid = (x-bar, y-bar)

x-bar = M_y/M

y-bar = M_x/M

M = ∫ρ[f(x) - g(x)]dx

M_y = ∫ρ(x)[f(x) - g(x)]dx

M_x = ∫(1/2)ρ[{f(x)}^2 - {g(x)}^2]dx

The Attempt at a Solution

First, I define y = x to be f(x), and y = 4x - x^2 to be g(x), so that the x^2 value will be positive, and simplify calculations. This leaves f(x) - g(x) as x^2 - 3x. The curves meet at x = 0 and x = 3, so I am integrating from 0 to 3.

Using the first integral for M, I get (27/2)ρ. This isn't a problem, a value along these lines was expecting.

The second integral is where the problem lays.

∫ρ(x)[f(x) - g(x)]dx simplifies to ρ(∫x^3 - 3x^2 dx) on the interval 0 to 3. The anti-derivative generated is: ρ([x^4]/4 - x^3)|(3-0)

This appears to simplify to -6.75, but since the curves meet, and thus the region for the centroid I am finding is contained entirely in the first quadrant, it seems that it's impossible for there to be a negative value in any of these integrals.

 

[haruspex]

First, I define y = x to be f(x), and y = 4x - x^2 to be g(x), so that the x^2 value will be positive,

And what will the value of f(x)-g(x) be at, say, x = 1?