Posted by **travis** on Sunday, November 28, 2010 at 10:35pm.

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

- calculus -
**MathMate**, Monday, November 29, 2010 at 9:45am
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)

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