To solve this problem, we must first define our given function: y = 6x^2 + 6x. The region of interest is bounded by this function, the x-axis (y=0), and a vertical line at x=3.
The centroid of a region is given by the equations:
x_bar = (1/A) * integral(x * f(x) dx)
y_bar = (1/2 * A) * integral(f(x)^2 dx)
where A is the area under the curve, and f(x) is the bounding function.
First, we need to calculate the area under the curve from x=0 to x=3. We find this with the integral of our function over this interval (A = integral( 6x^2 + 6x dx from 0 to 3)). The result of this computation is A = 81.
Next, we compute the x coordinate of the centroid (x_bar). This is accomplished by multiplying the x values by the function values (x * f(x) = x * (6x^2 + 6x)), and then integrating that from 0 to 3. Finally, we divide by the total area (81). The result, x_bar, is 13/6.
Computing the y coordinate of the centroid (y_bar) is similar, but now we need to square the function values (f(x)^2 = (6x^2 + 6x)^2) before integrating from x=0 to x=3. Again, we divide by twice the total area, which was found to be 81. After direct calculation, we find y_bar to be 109/5.
To summarize, the coordinates of the centroid of the region under the curve y = 6x^2 + 6x, from x=0 to x=3 is (x_bar, y_bar) = (13/6, 109/5).