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1. Determine the intersection points of the curves:
- Solve y = x^2 and y + 2x = 8 simultaneously to find the intersection points.
- Substituting y = x^2 into y + 2x = 8 gives x^2 + 2x = 8.
- Rearrange the equation to x^2 + 2x - 8 = 0 and solve for x to find the x-coordinates of the intersection points.
2. Set up the integral for the centroid:
- The formula for the centroid of a solid of revolution about the y-axis is given by:
\bar{x} = \frac{\int_{a}^{b} x*f(x) dx}{\int_{a}^{b} f(x) dx}
- In this case, f(x) represents the radius of the solid at a distance x from the y-axis.
3. Calculate the centroid:
- Integrate the x-coordinate of the centroid with respect to x over the bounds of the region to find the centroid.