Finding centroid of an area

Question

why should the integral be multiplied by 1/2 for finding the y bar? I dont understand that.

Thanks!

Answer 1

Cut it into small vertical rectangles each with height $y$ and length $dx$. Then the center of each rectangle is at $y/2$.

So the $y$ coordinate of the center is at $$Y=\frac{\int \frac{y}{2}ydx}{\int y dx}$$

Answer 2

The centroid is defined as the average of all points within the area. The vertical component is then defined by

$$Y=\frac{\iint y~dy~dx}{\iint dy~dx}=\frac{1}{2}\frac{\int y^2~dx}{\int y~dx}$$

Similarly, the $x$ component is given by

$$X=\frac{\iint x~dy~dx}{\iint dy~dx}=\frac{\int xy~dx}{\int y~dx}$$