# Homework Help: Simplify definite integral formula

1. Jan 2, 2012

### jvo

1. The problem statement, all variables and given/known data
Working my way through Zienkiewicz's "FEM its basis & fundamentals" I am stuck on the following example:
$- \int_{y_1}^{y_2} t N1 \left\{ \begin{array}{c} \sigma_x \\ \tau_y \end{array} \right\} dy = -\left\{ \begin{array}{c} k_x t (2 y_1 + y_2) (y_2 - y_1) / 6 \\ 0 \end{array} \right\}$

with values:
$N1 = \frac{ y_2 - y }{ y_2 - y_1 }$
$\sigma_x = k_x y$
and $\tau_y = 0$

2. Relevant equations
n/a

3. The attempt at a solution
My result for the above integral is: $k_x t \frac{ y_2^3 - 3 y_2 y_1^2 + 2 y_1^3 }{6 (y_2 - y_1)}$

Tried to simplify the formula by dividing by (y2 - y1) and (y2-y1)^2 but I don't see how to get to the same result as in the book. What am I doing wrong here?
For more detail, see the attachments with the page from the book and my results.
Jo

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2. Jan 2, 2012

### Dick

It is the same thing as in the book. $y_2^3 - 3 y_2 y_1^2 + 2 y_1^3=(2 y_1 + y_2) (y_2 - y_1)^2$. You just have to figure out how to factor the numerator.

3. Jan 3, 2012

### jvo

Thanks for your reply but I still don't see how to get there. Can you give a hint on how to approach this? As said in my post I tried long division by $(y_2 - y_1)$ and $(y_2 - y_1)^2$ but don't get even near your factored result.
I know factoring quadratics and checked 'recipes' as given here: http://www.analyzemath.com/polynomials/factor_polynomials.html
It's selfstudy, not homework. I realise missing something essential here, but what is it?
Tia,
Jo

PS this is a scan of some long division trials: http://utopia.knoware.nl/~jovo/images/Scan_img_026.jpg

Last edited: Jan 3, 2012
4. Jan 3, 2012

### Dick

You started off ok on the first trial. $y_2^2$ is good for the first term. Now keep the rest ordered in decending powers of $y_2$. So what you bring down after the first subtraction is $y_2^2 y_1 - 3 y_2 y_1^2$. Your next term in the quotient should be dividing $y_2$ into $y_2^2 y_1$ (the higher $y_2$ power term). Or $y_2 y_1$. Try it that way. It does divide evenly.

Last edited: Jan 3, 2012
5. Jan 3, 2012

### jvo

I got it! Thank you for your excellent help.
My new year's resolution is to practice factoring more :-)
Regards,
Jo