How Do You Calculate the Centroid of an Integrated Area?

[jennyp]

If I have an integrated area such as the blue area in the link below, what function can be written to find the location on the x-axis where half of the area is one side and half is on the other or more specifically a function that determines the x-axis location of the centroid?

http://images.encydia.com/thumb/f/f9/Areabetweentwographs.svg/180px-Areabetweentwographs.svg.png

 

[SteamKing]

If I have an integrated area such as the blue area in the link below, what function can be written to find the location on the x-axis where half of the area is one side and half is on the other or more specifically a function that determines the x-axis location of the centroid?

http://images.encydia.com/thumb/f/f9/Areabetweentwographs.svg/180px-Areabetweentwographs.svg.png

It depends on what exactly you want to find. The centroid of an area is not necessarily the same as the line which divides a region into two equal areas.

The location of the x coordinate of the centroid is the first moment of area about the y-axis divided by the area, or

x-bar = [ ∫ x * f(x) dx ] / [ ∫ f(x) dx ]