keywords: interval division, Riemann sums, limit of a Riemann sum
Integral Calculus is the subject that arose from the problem of trying to find the area of a region with a curved boundary. In general, this is calculated by a limiting process that yields gradually better approximations to the value.
Let $f$ be a function defined on the closed interval $[a,b]$. Take points $x_0, x_1, x_2, …, x_n$ such that
$a = x_0 < x_1 < x_2 < … < x_{n−1} < x_n = b, $
and in each subinterval $[x_i, x_{i+1}]$ take a point $c_i$.
The sum $f(c_0)(x_1 − x_0) + f(c_1)(x_2 − x_1) + … + f(c_{n−1})(x_n − x_{n−1})$ is called a Riemann sum for $f$ over $[a, b]$. Geometrically, it gives the sum of the areas of $n$ rectangles, and is an approximation to the area under the curve $y = f(x)$ between $x = a$ and $x = b$.
The (Riemann) integral of $f$ over $[a, b]$ is defined to be the limit $I$ (in a sense that needs more clarification than can be given here) of such a Riemann sum as $n$, the number of points, increases and the size of the subintervals gets smaller.
see, CHRISTOPHER CLAPHAM, JAMES NICHOLSON: The Concise Oxford Dictionary of Mathematics
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