Definition of Definite Integrals - Riemann Sums

In calculus, the integral of a function \( f(x) \) over an interval \( [a,b] \) is defined as a limit of Riemann sums.


Riemann sums

Let's divide the interval \( [a,b] \) into $n$ subintervals, each of width \( \; \Delta x = \dfrac{b-a}{n} \). We can then choose a point \( x_i \) in each subinterval \( [x_{i-1}, x_i] \) for \( i = 1, 2, \ldots, n \). The Riemann sum associated with this partition and choice of sample points is given by: \[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \]


The Integral as a Limit

The limit of the Riemann sum, as \( n \) approaches infinity, is defined as the definite integral of \( f(x) \) over \( [a,b] \), denoted by: \[ \int_a^b f(x) \; dx = \lim_{n\to\infty} R_n = \lim_{n\to\infty} \sum_{i=1}^{n} f(x_i) \Delta x \]



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