Final answer:
To sketch the region bounded by the curves, find the points of intersection and plot both functions with shading between them. The centroid's coordinates can then be calculated using integrals, where is the area of the region.
Step-by-step explanation:
Sketching the Region and Finding the Centroid
To sketch the region bounded by the curves y=x²-2x and y=5x-x², we first need to find the points of intersection by setting the two equations equal to each other:
\[x² - 2x = 5x - x²\]
Solving this gives us the points of intersection, which will define the limits of integration when we determine the centroid. The area between these curves can be visualized by plotting both functions on the same graph and shading the region between them.
Next, to find the centroid (\(\bar{x}\), \(\bar{y}\)) of the region, we will use these formulas:
\[\bar{x} = \frac{1}{A}\int_{a}^{b} x[A(x)]dx\]
\[\bar{y} = \frac{1}{2A}\int_{a}^{b} [y_1²(x) - y_2²(x)]dx\]
where \(y_1 = 5x - x²\), \(y_2 = x²-2x\), and \(A\) is the area between the curves.
Once the centroid coordinates are calculated, this point can be marked on the graph as the centroid of the shaded region.