Get same numerical and analytical integral's results?

Questions
1

Good morning-Afternoon.

I was recently programing an integral and decided to check how it would work compared to a numerical aproximation. I used Simpson and trapezoid rules. However, my results are several orders of magnitude different. I have attached my code.

Can someone please give me some hindsight on what I am doing wrong?

Thank you very much.

line_integral_attempt.nb (48.8 KB)

2

There are a number of ways to improve on this. Most important error is to not multiply trapezoid heights by step length. The value below is pretty good for an approximation.

step*(Total[Map[gausiana, N[PuntosEnsayo[[2 ;; -2]]]]] + 
   1/2 (gausiana[PuntosEnsayo[[1]]] + gausiana[PuntosEnsayo[[-1]]]))

(* Out[192]= 0.0000718038661611 *)
3

I see. Thank you bery much for the information. I am afraid I am not fluent enough in mathematica to propperly understand your proposed aproximation, but I will try to comprehend it.

4

Jaime,

your approach is basically correct. But you give the number of a point to the gaussian and not the x-value, which is in fact

PuntosEnsayo[[i]]

and not i. i is the number of your x-value. And you start with i =0, which is problematic for Tables.

Try in your code

For[i = 1, i < numero, i++, 
  valorNumerico = 
   valorNumerico + 
    step ((1/2)*(gausiana[PuntosEnsayo[[i]]] + 
         gausiana[PuntosEnsayo[[i + 1]]]))];
For[i = 1, i < numero, i++, 
  valorNumericoSimpson = 
   valorNumericoSimpson + 
    step (1/6)*(gausiana[PuntosEnsayo[[i]]] + 
       gausiana[PuntosEnsayo[[i + 1]]] + 
       4*gausiana[PuntosEnsayoMedios[[i]]])];
valorNumerico
valorNumericoSimpson

Daniel’s approach is Mathematica-like (avoid Do’s, For’s and so on if possible) and elegant.

He applies gausiana to each x-value, meaning each value in PuntosEnsayo. This is done with the Map-command. Then he adds all these values ( Total ) and finally multiplies with your step-width.
You can achieve this as well with

step*Total[gausiana /@ PuntosEnsayo]

neglecting the contributions at the ends of the interval

line_integral_attempt.nb (34.7 KB)

5

Yes, this is exactly what I was looking for.
Thank you very much for your help.