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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: $\int_a^b f(x) \; dx = \lim_{n\to\infty} R_n = \lim_{n\to\infty} \sum_{i=1}^{n} f(x_i) \Delta x$

Links and References

integrals and their applications in calculus.
Calculate integrals of functions .