∫ a b f ( x ) d x = lim ∑ f ( ξ i ) ( x i − x i − 1 ) int_a^bf(x)dx=limsum f(xi_i)(x_i-x_{i-1}) ∫abf(x)dx=lim∑f(ξi)(xi−xi−1)
来看一道利用积分和式求极限的公式：
设 f ( x ) f(x) f(x) 在 [0, 1] 上连续， u n = 1 n ∑ i = 1 n f ( i n ) u_n=frac1nsumlimits_{i=1}^nfleft(frac in ight) un=n1i=1∑nf(ni) 或 u n = 1 n ∑ i = 0 n − 1 f ( i n ) u_n=frac1nsumlimits_{i=0}^{n-1}fleft(frac in ight) un=n1i=0∑n−1f(ni)（0-1分为N份），则：
lim n → ∞ u n = lim n → ∞ 1 n ∑ i = 1 n f ( i n ) = ∫ 0 1 f ( x ) d x lim_{n o infty}u_n=lim_{n o infty}frac1nsumlimits_{i=1}^nfleft(frac in ight)=int_{0}^1fleft(x ight)dx n→∞limun=n→∞limn1i=1∑nf(ni)=∫01f(x)dx