In calculus, the integral of a function \( f(x) \) over an interval \( [a, b] \) is defined as a limit of Riemann sums.
In what follows, \( C \) represents the constant of integration where applicable.
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 sample 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:
\[ \int_a^b f(x) \; dx = \lim_{n\to\infty} R_n = \lim_{n\to\infty} \sum_{i=1}^{n} f(x_i) \Delta x \]