1. Center of gravity

I've already taken calculus but I forgot how to a problem like this and I don't have my book anymore.

I have to find the center of gravity of a parallelogram with coordinates:
(0,0), (1,1), (3,1), (2,0) using calculus. I think I need to use integrals, but I'm hoping there is a formula. Any advice? Thanks.

2. Originally Posted by jzellt
I've already taken calculus but I forgot how to a problem like this and I don't have my book anymore.

I have to find the center of gravity of a parallelogram with coordinates:
(0,0), (1,1), (3,1), (2,0) using calculus. I think I need to use integrals, but I'm hoping there is a formula. Any advice? Thanks.
Just wondering. Is there a density function given? Or something that describes the density of the lamina that we are finding the center of gravity?

If so, then there are two formulas for the coordinates of the center of gravity:

$\displaystyle \bar{x}=\frac{\displaystyle\iint\limits_R x\delta\left(x,y\right)\,dA}{\displaystyle\iint\li mits_R \delta\left(x,y\right)\,dA}$

$\displaystyle \bar{y}=\frac{\displaystyle\iint\limits_R y\delta\left(x,y\right)\,dA}{\displaystyle\iint\li mits_R \delta\left(x,y\right)\,dA}$

However, if there is no information on the density, I would probably assume it to be constant everywhere. This will give us a special case of the center of gravity. Treating $\displaystyle \delta\left(x,y\right)$ as a constant, we see now that

$\displaystyle \bar{x}=\frac{\delta\left(x,y\right)\displaystyle\ iint\limits_R x\,dA}{\delta\left(x,y\right)\displaystyle\iint\li mits_R \,dA}=\frac{\displaystyle\iint\limits_R x\,dA}{\displaystyle\iint\limits_R \,dA}$

$\displaystyle \bar{y}=\frac{\delta\left(x,y\right)\displaystyle\ iint\limits_R y\,dA}{\delta\left(x,y\right)\displaystyle\iint\li mits_R \,dA}=\frac{\displaystyle\iint\limits_R y\,dA}{\displaystyle\iint\limits_R \,dA}$

This gives us the coordinates of the centroid.

Does this help?

--Chris

3. All that was given is what I posted. Thanks for the equations.

I guess it's been a long time...but, how do I take the double integral of points? Do I find an equation between the two points first? A little more help would be appreciated. Thanks.

4. Originally Posted by jzellt
All that was given is what I posted. Thanks for the equations.

I guess it's been a long time...but, how do I take the double integral of points? Do I find an equation between the two points first? A little more help would be appreciated. Thanks.
You need to find the equations of the lines that construct the sides of the parallelogram. Once you have those, I would suggest plotting the region. Then you need to determine the proper integration limits.

From the looks of it, you will be using the centroid equation I gave you. Keep in mind that the denominator value in both $\displaystyle \bar{x}$ and $\displaystyle \bar{y}$ is the area of the parallelogram.

Can you try to take it from here?

--Chris