Locating ȳ of area, using calculus

 

Statics, Chapter 9 - Center of Gravity and Centroid

Consider the area in (Figure 1). (Figure 1).

Locate the centroid ˉy of the area. 

Express your answer to three significant figures and include the appropriate units.

 STEP 1  

Consider the equation for the area of the differential element (dA), and that its equation is: 

dA = ydx, or 

dA = xdy 

(in this instance since we will use dA = ydx since we are given the equation for y.)

Consider the equation for a centroid of the y co-ordinate as well. It is:

ˉy = y/2

 

 STEP 2  

Now, we will use the formula for the center of a side length (y-bar) to calculate the ˉy using the centroid formula.

dA =  ydx or (1-1/4x^2), and ˉy = y/2 or 1/2*(1-1/4x^2)

The integral is taken on the area range -2 to 2.

 STEP 3  

All that is left to do is anti-differentiate.

 

and simplify.

Answer is (2/5) m, or 0.400 m

Statics, Chapte