Original
The process of approximating areas under curves led to the notion of a Riemann sum
{{MATH_7}}
where {{MATH_8}} is a nonnegative, continuous function on the interval {{MATH_9}}, and
{{MATH_10}} is a sample point for the {{MATH_11}} rectangle, {{MATH_12}}.
The limit of Riemann sums, as {{MATH_13}}, gives the exact area between the curve {{MATH_14}} and the interval on the {{MATH_15}}axis:
{{MATH_16}}
It does not matter whether we consider only right Riemann sums, or left Riemann sums, or midpoint Riemann sums, or others: the limit of any kind of Riemann sum as {{MATH_17}} is equal to the area, as long as {{MATH_18}} is nonnegative and continuous on {{MATH_19}}.