Formal Definition of the Integral

The most standard definition is the Riemann integral.

Let \( f:[a,b] \to \mathbb{R} \) be a bounded function on a closed interval \([a,b]\).

1. Divide \([a,b]\) into \(n\) subintervals: \[ a = x_0 < x_1 < x_2 < \cdots < x_n = b \] This collection of points is called a partition \(P\).
2. On each subinterval \([x_{i-1},x_i]\), choose a sample point \(\xi_i \in [x_{i-1},x_i]\).
3. Form the Riemann sum: \[ S(P,f) = \sum_{i=1}^n f(\xi_i)(x_i - x_{i-1}) \]
4. Take the limit as the partition gets finer (the largest subinterval length goes to 0): \[ \int_a^b f(x)\,dx = \lim_{\|P\|\to 0} \sum_{i=1}^n f(\xi_i)(x_i - x_{i-1}) \] \[ \int_a^b f(x)\,dx = \lim_{\|P\|\to 0} \sum_{i=1}^n \color{orange}f(\xi_i)\color{purple}\Delta x_i \] \[ \color{orange}f(\xi_i) = \text{height} \] \[ \color{purple}\Delta x_i = \text{width} \] \[ \|P\| \;=\; \max_{1 \le i \le n} \Delta x_i. \] max is the largest subinterval length.

Intuition

The partition breaks the interval into small pieces. The sum adds up "height × width" rectangles (\(f(\xi_i)\) is the height, \((x_i-x_{i-1})\) is the width). The limit makes the rectangles infinitely thin, giving the exact area or accumulation.