A-Level Maths: The Trapezium Rule for Approximating Integrals | A-Level 数学：梯形法则近似计算积分

📚 A-Level Maths: The Trapezium Rule for Approximating Integrals | A-Level 数学：梯形法则近似计算积分

The trapezium rule is a standard numerical method used to estimate the value of a definite integral. In A-Level Mathematics, it is especially useful when the function cannot be integrated easily or when the curve is only given as a set of data points.

梯形法则是估算定积分值的标准数值方法。在 A-Level 数学中，当函数难以直接积分，或者曲线仅以一组数据点给定时，它特别有用。

1. What Is the Trapezium Rule? | 什么是梯形法则

The trapezium rule divides the area under a curve into a number of parallel vertical strips. Each strip is treated as a trapezium, and the total area of these trapeziums is used as an approximation for the definite integral.

梯形法则将曲线下方的区域分成若干条垂直的窄条。每一条都被看作一个梯形，这些梯形面积之和就被用作定积分的近似值。

This method is sometimes called the trapezoidal rule. In British A-Level courses, the name “trapezium rule” is more common.

这种方法有时也被称为梯形求积法则。在英国 A-Level 课程中，更常使用“trapezium rule”这个名称。

2. Why Do We Need It? | 为什么需要近似计算积分

Some important functions do not have antiderivatives that can be written using elementary functions. For example, the integral of eˣ² from 0 to 1 cannot be expressed using simple algebraic, trigonometric, or exponential functions.

有些重要的函数没有可以用初等函数表示的原函数。例如，eˣ² 在 0 到 1 上的积分就无法用简单的代数、三角或指数函数来表达。

In other situations, we may only know the value of y at certain x-values because the data comes from an experiment. In both cases, numerical methods such as the trapezium rule allow us to obtain a useful approximation.

在其他情况下，我们可能只知道某些 x 值对应的 y 值，因为这些数据来自实验。在这两种情况下，像梯形法则这样的数值方法都可以帮助我们得到有用的近似值。

3. The Formula and Notation | 公式与记号

Suppose we want to approximate the integral of f(x) from x = a to x = b. We divide the interval [a, b] into n equal strips, each of width h.

假设我们要近似计算 f(x) 从 x = a 到 x = b 的积分。我们将区间 [a, b] 分成 n 个等宽的窄条，每个窄条的宽度为 h。

h = (b − a) / n

The x-values are x₀ = a, x₁ = a + h, x₂ = a + 2h, and so on, up to x

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