Can I use experimental data to solve equations with integrals?

[rakue]

Hi, I don't know if this is the proper part of the forum to ask this, but I'm trying to figure out how I can obtain a certain value from an equation that contains an integral if I want to use experimental data. To keep it simple, the equation looks like this:

B= (∫G(f)df)² / (∫G(f)²df)

where it is to be evaluated from 0 to ∞.

But to keep my question simple, say, my equation is: B= ∫G(f)df

Now, I have experimental values for "G(f)" and it's corresponding "f" (I'm not even sure if I'm understanding this correctly).

Is it right for me to simply, multiply the G's with their corresponding f's and then take the sum? So in effect, I'm basically doing this:

Ʃ(G_i*f_i) where i is the index for each value of f

Am I still integrating if I do this? If not, what's the proper approach to this?

Thank you very much for all your help.

 

[SteamKing]

Based on the equations you have written, why not plot G(f) v. f and [G(f)]^2 v. f ?

If these plots turn out to be smooth curves, there are a number of numerical integration techniques which you can use to calculate B.

 

[bossman27]

Is it right for me to simply, multiply the G's with their corresponding f's and then take the sum? So in effect, I'm basically doing this:

Ʃ(G_i*f_i) where i is the index for each value of f

Am I still integrating if I do this? If not, what's the proper approach to this?

What do you mean by the index for each value of f? I think what you're referring to is incorrect. Numerical integration techniques basically rely on one or another method for calculating the area under a curve, so you need some measure of "width."

In the simplest case, $\int G(x) dx$ could be approximated by $\sum{G(x_{i}) \Delta x}$ where $\Delta x$ is an appropriate measure of the "width" of a column whose height is determined by $G(x_{i})$. Usually this would just be: $\Delta x = (x_{i+1}-x_{i})$ Graphically, it would look like approximating the area under the curve with a set of vertical bars (this is a pretty common teaching tool for basic calculus: the actual integral is basically the limit as $\Delta x \rightarrow dx$, at least when we're talking about a Riemann integral).

With that said, that method is very rudimentary (although that might be what you're looking for, depending on the context of your question) and is rarely, if ever, used in actual numerical computations or data analysis. If you're using any kind of data analysis software (i.e. MATLAB) there are much more accurate techniques for numerical integration.