200 Definite Integral
A scalar defined as the limit of a sum of function values on subintervals that is used to measure accumulated size along an interval.
definition [d] (Definite Integral) The number obtained from a function \(f\) on \([a, b]\) by partitioning \([a, b]\) into \(n\) subintervals of equal width \(\Delta x = (b-a)/n\), choosing sample points \(x_{i}^{*}\) in each, and taking the limit of Riemann sums:
- \(\displaystyle \int_{a}^{b} f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_{i}^{*})\, \Delta x\) ,
provided the limit exists and is independent of the choice of sample points.
where
- \(a\) is the lower limit; \(b\) is the upper limit.
- \(\Delta x = (b-a)/n\) is the subinterval width.
- \(x_{i}^{*}\) is a sample point in the \(i\)th subinterval \([x_{i-1},\, x_{i}]\).
- \(f\) is integrable on \([a,b]\) when this limit exists.