find the centroid of the region bounded by x^2-12x and the x-axis

Answer 1

The expression x²-12x crosses the x-axis twice, at x=0 and x=12.

The curve stays below the x-axis on the interval [0,12].

See:
http://img207.imageshack.us/img207/3690/1291001724centroid.png

The area is therefore:
I1=∫(x^2-12x)dx from x=0 to x=12.
By symmetry, the centroid lies on the line x=6.
To find the y-distance, evaluate the integral in which each slice is multiplied by y/2=(x^2-12x), equal to the centroid of each slice:
I2=∫(1/2)(x^2-12x)²dx
The y-position of the centroid is then:
yc=I2/I1

I get yc=-14.4 (below the x-axis)