Question

As shown, a section of a roller coaster track (figure 3) follows the form of y=(2/3)x1.5. locate the centroid x? of the track between x=0 m and x=3.5m

Answer 1

F(x)=(2/3)x^1.5

The centroid position along the x-axis can be obtained by 
integrating the function * x to get the moment about the y-axis, 
then divide by the area of the graph,
all between x=0 to x=3.5m.

Expressed mathematically,
x_bar=(∫f(x)*x dx )/(∫ f(x) dx  limits are between x=0 and x=3.5m
=15.278 m^3 / 6.1113 m^2
=2.500 m