find the centroid of the region bounded by x^2-12x and the x-axis
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)
To find the centroid of the region bounded by the curve x^2-12x and the x-axis, we need to first determine the coordinates of the points where the curve intersects the x-axis. The x-axis is given by the equation y = 0.
Setting x^2-12x = 0, we can factor out an x from the equation to get x(x-12) = 0. This equation is satisfied when either x = 0 or x - 12 = 0.
So, the curve intersects the x-axis at x = 0 and x = 12.
To find the y-coordinate of the centroid, we need to compute the area of the region, which is defined as the integral from x = 0 to x = 12 of the function x^2-12x.
The integral of the function x^2-12x with respect to x is given by ∫(x^2-12x) dx = (1/3)x^3 - 6x^2.
Using the fundamental theorem of calculus, we can evaluate this integral from x = 0 to x = 12:
∫[(1/3)x^3 - 6x^2] dx = [(1/3)(12)^3 - 6(12)^2] - [(1/3)(0)^3 - 6(0)^2] = 288 - 864 - 0 + 0 = -576.
The negative sign is because the region lies below the x-axis.
To find the y-coordinate of the centroid (Cy), we divide the integral result by the area:
Cy = (-576) / (12 - 0) = -576 / 12 = -48.
So, the y-coordinate of the centroid is -48.
The x-coordinate of the centroid (Cx) of the region is given by the formula:
Cx = ∫[x(x^2-12x)] dx / ∫[(x^2-12x)] dx.
Evaluating these integrals:
∫[x(x^2-12x)] dx = [(1/4)x^4 - 6(x^3)] = (1/4)(12)^4 - 6(12)^3 = (1/4)(20736) - 6(1728) = 5184 - 10368 = -5184.
∫[(x^2-12x)] dx = (1/3)(12)^3 - 6(12)^2 = 288 - 864 = -576.
Cx = (-5184) / (-576) = 9.
Therefore, the centroid of the region bounded by x^2-12x and the x-axis is at the point (9, -48).