Second moment of area - Need walkthrough

Hi all,

I'm at uni starting off engineering, and I'm looking for a walk through in this algebra involved in the second moment of area. While I know it's probably pretty basic I'm undertaking a bridging course to try and keep up with my maths.

http://home.exetel.com.au/peleus/smoa.jpg

Here's a picture of the most relevant lecture slide discussing the problem.

I'll type out the steps they undertook to get the final formula for the second moment of area for a rectangle.

On the next page we take the integral of this, which I can do fine.

This gives

1. $\displaystyle I = \frac{b}{3}[y^3]$ with limits +h/2 and -h/2

Taking it further we end up with

2. $\displaystyle I = \frac{b}{3}[\frac{h^3}{8}-(-\frac{h^3}{8})]$

and finally we take it to the step

$\displaystyle I = \frac{bh^3}{12}$

Ok, I can understand a bit about this but here are my questions.

- Why are the limits h/2 and -h/2, isn't this simply the middle of the rectangle?

- Why is $\displaystyle dI = y^2 dA$, where does the $\displaystyle y^2$ come from?

- I understand that the $\displaystyle \int y^2$ is $\displaystyle \frac{1}{3}y^3$, but where / why does the b jump in with step 1? why are we multiplying it by b?

Any help is greatly appreciated.