Help understanding the mathematical expression of product of Inertia

[superjose]

Homework Statement

Hello! I'm stuck and confused. I've been browsing the web, and reading books but can't seem to get it right. I've been looking for the mathematical expression for product of Inertia. I've managed to understand the 50% of it. But I'm stuck on the first part.

Homework Equations

Equation:

I_xy = I_x'y' + AX*Y*

Where A = Area of the figure
X* = The center of gravity in "x" axis
Y* = The center of gravity in "y" axis


But I can't figure what's I_x'y'. I know that if the figure is symmetrical it will give 0. But how is that possible?

Please note that I'm trying to solve problems without integration. I've been given real values.

Thanks a lot in advance.

The Attempt at a Solution

 

[jbsteele]

Ix'y' represents the product of inertia at a given point (x', y') relative to the center of gravity of the figure. This is calculated by taking the moment of inertia for each axis about the center of gravity and multiplying it together. For example, for a rectangle of dimensions L x W, the product of inertia would be:Ix'y' = LW3/12 - (LW2/4)*(L/2)*(W/2) where L is the length of the rectangle and W is the width. The first term is the moment of inertia of the rectangle about the x-axis and the second term is the moment of inertia of the rectangle about the y-axis. The total product of inertia can then be found by adding Ix'y' to the product of the area of the figure and the distance between the center of gravity and the point (x', y'):Ixy = Ix'y' + AX*Y* Hope this helps!