Definition:Riemann Integral/Historical Note

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Historical Note on Riemann Integral

Consider the Riemann sum:

$\displaystyle \sum_{i \mathop = 1}^n \ f \left({c_i}\right) \ \Delta x_i$

Historically, the definite integral was an extension of this type of summation such that:

The finite difference $\Delta x$ is instead the infinitely small difference $\rd x$
The finite sum $\Sigma$ is instead the sum of an infinite number of infinitely small quantities: $\int$

Hence the similarity in notation:

\(\displaystyle \sum_a^b \ f \left({x}\right) \, \Delta x\) \(\to\) \(\displaystyle \int_a^b f \left({x}\right) \rd x\) $\quad$ as $\Delta x \to \rd x$ $\quad$

The notion of "infinitely small" does not exist in the modern formulation of real numbers. Nevertheless, this idea is sometimes used as an informal interpretation of the definite integral.

Riemann established this definition in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe of $1854$, on the subject of Fourier series.