Riemann Sum

Core Sentence / 核心句

A Riemann sum approximates continuous accumulation by adding products of sampled heights and small interval widths.

中文理解：先用许多小矩形近似局部贡献，再让最宽区间趋近零，得到精确积分。

\int_{a}^{b} f (x) d x = m a x Δ x_{i} \to 0 lim \sum f (x_{i}^{*}) Δ x_{i}

For $a = x_{0} < \dots < x_{n} = b$ :

L_{n} = i = 1 \sum n f (x_{i - 1}) Δ x_{i}, R_{n} = i = 1 \sum n f (x_{i}) Δ x_{i},

M_{n} = i = 1 \sum n f (\frac{x _{i - 1} + x _{i}}{2}) Δ x_{i},

T_{n} = i = 1 \sum n \frac{f ( x _{i - 1} ) + f ( x _{i} )}{2} Δ x_{i} .

Equal subintervals use $Δ x = (b - a) / n$ ; unequal partitions use each interval’s own $Δ x_{i}$ .