Is it Possible to Model Functions From Exchange Rates? Take for instance the Euro to Dollar exchange rate (EURUSD). If I were to solely consider the monthly exchange rates over the interval [a,b] where:
a=02/01/18
b=07/01/18
Is it possible to model a function whereby I can then use to determine the average value of this function over this interval? I'd ultimately integrate this function to find its average-value using:
$$1/5\int_0^5 F(x)dx = Yav$$
Where F(x) represents the modeled function over interval [a,b], and Yav represents the average value.
I usually determine the current average monthly rate of an exchange rate by taking its previous 5 monthly candlesticks' high to low difference, adding them, then dividing the sum by 5. I feel this is the simple approach.
Is modeling such a function too overly complicated?
I'm still learning about integration, so please excuse any terms I might have gotten wrong. 
 A: The first step is to identify what your data on the exchange rate is: more likely than not, it will consist of a number of "measured" exchange rates $y_i$ at a number of specific times $t_i$.
As a result, you won't be able to directly use the definition of the integral for continuous functions you mention in your question, because you don't really know how the function behaves in the time intervals $(t_i,t_{i+1})$ between the time points you've obtained in your data.
But, you can guess!
The simplest thing you could do is assume that the exchange rate doesn't change between time points in your data. Luckily, we know the integral of a constant function;
$$\int_{t_i}^{t_{i+1}} y_i\ dt = y_i \Delta t_i$$
where $\Delta t_i$ is the time difference between $t_i$ and $t_{i+1}$. 
Therefore, you can do a bunch of these integrals, add them all up, and then divide by the total time difference $\Delta t$ to get an approximation for the average of $y$;
$$\bar{y} =\frac{1}{\Delta t}\sum_{i=1}^n y_i \Delta t_i$$
This seems sloppy though: clearly the exchange rate does change between your time points, you just don't know how. So you may decide to assume that it changes linearly between each time point, which would get you the following approximation if you follow the same procedure/logic as above:
$$\bar{y} =\frac{1}{\Delta t}\sum_{i=1}^n \frac{y_i + y_{i+1}}{2} \Delta t_i$$
This approximation is well-known as the trapezoidal rule, and has very nice bounds on its error among other useful properties. You can, of course, assume whatever specific form of change between time points you prefer, which will change the nature of your approximation.
Another avenue of inquiry you may pursue, but is probably not what you're looking for, consists of modeling the process as a "continuous" stochastic function defined at every point and then using stochastic calculus to find the integral (and consequently mean) of the function. Doing so, however, would imply you're guessing at the kind of stochastic processes the exchange rate evolution is, and is considerably trickier than the discrete methods shown above.
Hope this helps!
