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

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