Relationship between Riemann Sum and the Integral

[Flumpster]

Homework Statement

The notation for a Riemann sum - Ʃ f(x*i)Δx - is very similar to the notation for the integral (the Ʃ becomes ∫, the f(x*i) becomes f(x) and the Δx becomes dx).

$$\int f(x)dx = \lim_{n \to \infty}\sum_{k=0}^{n} f(x_i) Δx$$

Is there a way to explicitly define the values on the right hand side in terms of the left hand variables (for instance to define dx in terms of Δx)? Or is it just that the whole left side equals the right side and that's it?

Thank you.

 

[lippyka]

Homework EquationsThe equation given in the problem statement.The Attempt at a SolutionNo, it is not possible to explicitly define the values on the right hand side in terms of the left hand variables, as the left hand side represents the integral of the function, while the right hand side represents an approximation of the integral using a Riemann sum. The two sides of the equation are simply related by the fact that the integral is equal to the limit of the Riemann sum as the number of partitions increases (i.e. as n → ∞).