As you may or may not know, the theory of math integration started with what is called the Peano-Jordan measure.
In order to calculate the integral of a function, they started calculating the area subtended by the function.

The process is simple (and you can see the picture to understan it): let's say you want to calculate the area subtended by the function p(x) in a given interval [x₁,x₂]. All you have to do is divide this interval in a finite number of steps and build rectangles as shown in the picture. Since the function I'll give you is continuous because it is a polynomial function ( p(x) ) you are sure that the vertical lines of the rectangles are going to meet the function somewhere. In each interval you get multiple values of the function because each step is made by infinite points. For each step you can take either the max(p(x)), or min(p(x)) or an average value that can be either avg(x)=[max(p(x)+min(p(x))]/2 or the value that the function gets in the middle point of each step.
Depending on what value you take you get an area approximation which is a little bigger than the real area (taking max(p(x))), a little smaller than the real area (taking min(p(x))) or very near the real area (taking avg(p(x))).
Anyway, Riemann has proved that when the measure of each step becomes smaller and smaller (tending to 0) all these values tend to one value which is the correct area.

What you're asked to do here is to calculate an approximation of the area using a step length of 0.01 units
and taking the middle value to calculate the height of the rectangles. The middle value is the value that p(x) gets in the exact middle point of each step.
So, if for a given p(x) in a certain step you get min(p(x))=8, max(p(x))=10 and middle(p(x))=8.5, even if the average avg(p(x))=9 I want you to use 8.5.
The area of that rectangle will then be 8.5*0.01=0,085 units². The total area is the sum of each rectangle area you'll get from x₁ to x₂. If for example x₁=1 and x₂=4 you're going to calculate (4-1)/0.01=3*100=300 rectangles.

Note that if middle(p(x))<0 then p(x) in that point is negative and the corresponding rectangle will be negative too.
The polynomial I provide has this structure:

p(x) = (n0/d0) + (n1/d1)*x^1 + ... + (nk/dk)*x^k
(where "+" signs could be "-" either and where the first coefficient could be negative as well)

k is random calculated by my script and n_k, d_k are the random numerator and denominator of the k^th coefficient of p(x).
So, resuming:

1. Get p(x).
2. Get [x₁,x₂] (note that [a,b] mean that the interval is inclusive)
3. Divide [x₁,x₂] in 100*(x₂-x₁) steps
4. Sum up all the rectangles you get multiplying 0.01*middle(p(x)) for each step.
5. Now you get the approximation of the area.

When you have the correct approximation pick up the absolute value and round it to an integer then calculate the lowercase md5 of the number.
That will be the solution you have to pass to the solution page.

Example 1: 345.23 -> solution=md5(345)
Example 2: 123.89 -> solution=md5(124)
Example 3: -567.44 -> solution=md5(567)
Example 4: -567.50 -> solution=md5(568)

Notes: to calculate the result the script uses regular php vars.
To check php's floating point precision go to www.php.net
Click here to start the challenge

"http://bright-shadows.net/GDO/TBS/challenges//programming/peanojordan/solution.php?solution="+letters

The page will then tell you the password for this challenge.
Time limit is 4 seconds.

Good luck!
Sfabriz