微积分I（标准路径） · free preview

§5.6 Numerical integration

当被积函数不存在初等代数原函数，因而无法使用微积分第一基本定理时，我们如何准确计算诸如 $\int_0^1 e^{-x^2} \, dx$ 这样的定积分？是否有一些方法可以在黎曼和中不使用极大的 $n$ 值的情况下得到精确的估计？

1. Introduction

Introduction

The Trapezoid Rule

Comparing the Midpoint and Trapezoid Rules

Simpson's Rule

For $f(x) = \frac{1}{x^2}$ on $[1,2]$ , which is concave up, the trapezoid and midpoint rules satisfy

For $\int_0^3x^2\,dx$ with three equal subintervals, the left endpoint sum $L_3$ equals

Simpson's rule combines the midpoint and trapezoid estimates on matching subintervals as

A braking car's velocity table with three panels gives $L_3 = 165.6$ feet and $R_3 = 105.6$ feet. Compute the trapezoid estimate $T_3$ in feet.

With $M_3 = 143.4$ feet and $T_3 = 135.6$ feet for the braking car, compute Simpson's estimate $S_6$ in feet.

For the braking car whose velocity graph is concave down, the midpoint estimate $M_3 = 143.4$ feet is an overestimate of the true distance.

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