# Centroid of a 2D region integral explanation

I've been told that to find the center of mass for a region defined in the x-y plane I need to use this formula

Edit: A region defined parametrically by $x(t)$ and $y(t)$ where $t$ is $a \leq t \leq b$

$Y=\dfrac{\int_a^b y(t) x(t) y'(t)dt}{\int_a^b x(t) y'(t)dt}$

Where the denominator is the area of the region in question. I can almost see why it works but I'm hung up on the $y'(t)$

I can see how this

$Y=\lim\limits_{n \to \infty}\dfrac{\sum\limits_{i=0}^n y(t_i)x(t_i)}{\sum\limits_{i=0}^{n} x(t_i)}$

represents the y coordinate of the center of mass of the region but the problem is how it translates to integral form.

$\int_a^b y(t)x(t)y'(t)~dt = \lim\limits_{n \to \infty} \sum\limits_{x=0}^n y(t_i)x(t_i)(y(t_i +\Delta t)-y(t_i))$

and

$\int_a^b x(t)y'(t)~dt = \lim\limits_{n \to \infty} \sum\limits_{x=0}^n x(t_i)(y(t_i +\Delta t)-y(t_i))$

It looks like it would work just as long as the $y(t_i +\Delta t)-y(t_i)$ could cancel

When you divide the two you would get $$\dfrac{\int_a^b y(t)x(t)y'(t)~dt}{\int_a^b x(t)y'(t)~dt} \lim\limits_{n \to\infty} \dfrac{\sum\limits_{x=0}^n y(t_i)x(t_i)(y(t_i +\Delta t)-y(t_i))}{\sum\limits_{x=0}^n x(t_i)(y(t_i +\Delta t)-y(t_i))}$$

Where do I go from here to show that this is in fact the weighted average of the two functions?

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You should explain what $x,y$ and $t$ are. – nbubis Apr 25 '12 at 13:45