Example 7.4.1: The Trapezium Rule | 示例 7.4.1：梯形法则

📚 Example 7.4.1: The Trapezium Rule | 示例 7.4.1：梯形法则

matics example on numerical integration. It asks you to estimate the definite integral ∫₁³ √x dx using the trapezium rule with four strips, and then to judge how good the estimate is.

示例 7.4.1 是 AQA A-Level 纯数学中关于数值积分的核心例题。题目要求使用梯形法则、取 4 条，估计定积分 ∫₁³ √x dx 的值，并进一步判断这个估计的好坏。

This article works through the full solution, checks it against the exact value, explains the sign of the error, and collects the common exam errors and exam-style variations.

本文将完整讲解求解过程，用精确值加以检验，解释误差的正负，并汇总考试中的常见错误与常见变式。

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1. What Is Example 7.4.1? | 什么是示例 7.4.1？

to estimate the value of the integral from 1 to 3 of the square root of x, written as ∫₁³ √x dx.

在 AQA 教材中，示例 7.4.1 提出以下问题：使用梯形法则、取 4 条，估计从 1 到 3 的根号 x 的定积分，即 ∫₁³ √x dx 的值。

The function y = √x is continuous on the closed interval [1, 3], so the method can be applied directly without any special care about discontinuities.

函数 y = √x 在闭区间 [1, 3] 上连续，因此该方法可以直接使用，无需特别处理间断点。

The example is deliberately simple so that the exact value can be found afterwards by integration, which lets you measure the size and the sign of the numerical error.

这个例题被刻意设计得比较简单，因为之后可以用积分求出精确值，从而可以衡量数值误差的大小和正负。

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2. The Standard Form of the Question | 标准形式的问题

In AQA papers, a trapezium rule question usually gives the integrand f(x), the lower limit a, the upper limit b, and the number of strips n. Everything else has to be worked out.

在 AQA 试卷中，梯形法则题目通常会给出被积函数 f(x)、下限 a、上限 b 以及条数 n，其余内容需要自己推导。

Here a = 1, b = 3 and n = 4, so the strip width h is calculated from h = (b − a) / n.

这里 a = 1，b = 3，n = 4，因此条宽 h 由 h = (b − a) / n 计算。

h = (3 − 1) / 4 = 2 / 4 = 0.5

This width labels the x-coordinates x₀, x₁, x₂, x₃ and x₄, which are the equally spaced points at which the curve is sampled.

这个宽度用于标出 x 坐标 x₀、x₁、x₂、x₃ 和 x₄，也就是对曲线进行采样的等距点。

Notice the counting rule: n strips always produce n + 1 ordinates. Four strips therefore need five y-values, and this is the single most common source of lost marks.

注意计数规则：n 条必然对应 n + 1 个纵坐标。4 条因此需要 5 个 y 值，这也是最常见的失分点。

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3. Why Use the Trapezium Rule? | 为什么使用梯形法则

The trapezium rule is a numerical method used when an integral cannot be evaluated analytically, or when only a table of data points is known.

梯形法则是一种数值方法，用于积分无法解析求解，或者只知道一组数据点的情形。

It works by splitting the area under the curve into a number of trapezium-shaped strips and adding their areas, replacing the curve by a series of straight chords.

它的原理是把曲线下方的区域分割成若干个梯形条，用一系列直线弦代替曲线，再把这些梯形面积相加。

For y = √x on [1, 3], the second derivative is f″(x) = −1 / (4x^3/2 which is negative throughout the interval. The curve is therefore concave down on [1, 3].

对于 [1, 3] 上的 y = √x，二阶导数为 f″(x) = −1 / (4x^3/2)，在整个区间上恒为负。因此曲线在 [1, 3] 上是向下凹的。

Because each chord lies below the curve, every trapezium sits inside the true area, so the straight-line tops make the estimate too small. This is the geometric reason behind the sign of the error discussed later.

由于每条弦都位于曲线下方，每个梯形都落在真实面积之内，所以用直边代替曲边会使估计值偏小。这就是后文讨论误差正负时的几何原因。

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4. The Trapezium Rule Formula | 梯形法则公式

The version printed in the AQA formula booklet is written in terms of the ordinates y₀, y₁, …, yₙ.

AQA 公式册中给出的形式是用纵坐标 y₀、y₁、…、yₙ 表示的。

∫ₐᵇ f(x) dx ≈ ½h [ y₀ + yₙ + 2(y₁ + y₂ + … + y_n−1) ]

Here h = (b − a) / n is the width of each strip, y₀ = f(a) and yₙ = f(b).

其中 h = (b − a) / n 是每条的宽度，y₀ = f(a)，yₙ = f(b)。

The pattern to remember is simple: the two end ordinates are counted once, and every interior ordinate is counted twice because each interior vertical edge is shared by two neighbouring trapezia.

需要记住的规律很简单：两个端点处的纵坐标只算一次，而每个内部纵坐标都要算两次，因为每条内部竖直边被相邻两个梯形共用。

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e bracket must contain 1 + 1 + 2 + 2 + 2 ordinate copies, that is seven y-values in total.

一个实用的自检方法是数一数每个 y 值在括号中出现几次。当 n = 4 时，括号中应包含 1 + 1 + 2 + 2 + 2 份纵坐标，即总共七个 y 值。

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5. Building the Table of Values | 建立数值表

The first working mark in an exam is almost always for a correct table of ordinates, so write it out clearly before substituting into the formula.

考试中第一个过程分几乎总是给正确的纵坐标表，因此务必先清楚地写出来，再代入公式。

x	1	1.5	2	2.5	3
y = √x	1	1.2247449	1.4142136	1.5811388	1.7320508

5, x₂ = 2, x₃ = 2.5 and x₄ = 3, obtained by adding h = 0.5 repeatedly starting from a = 1.

x 值为 x₀ = 1、x₁ = 1.5、x₂ = 2、x₃ = 2.5 和 x₄ = 3，是从 a = 1 出发反复加 h = 0.5 得到的。

Keep at least six decimal places in the table. Rounding the ordinates to two decimal places at this stage can shift the final answer by several units in the third decimal place.

表格中至少保留六位小数。若在此阶段就把纵坐标四舍五入到两位小数，最终答案的第三位小数可能会偏差好几个单位。

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6. Step-by-Step Solution | 逐步求解

Substitute the five ordinates into the bracket, taking care to double only the interior values.

把这五个纵坐标代入括号，注意只把内部的值加倍。

S = y₀ + y₄ + 2(y₁ + y₂ + y₃)

S = 1 + 1.7320508 + 2(1.2247449 + 1.4142136 + 1.5811388)

S = 2.7320508 + 2 × 4.2200973 = 11.1722453

The interior sum 4.2200973 is worth writing down separately, because examiners like to see the doubling step explicitly.

内部之和 4.2200973 值得单独写出，因为阅卷人希望看到加倍这一步被明确写出来。

Finally multiply by ½h = 0.25.

最后乘以 ½h = 0.25。

∫₁³ √x dx ≈ 0.25 × 11.1722453 = 2.7930613

So the trapezium rule gives approximately 2.7931, or 2.79 to three significant figures, or 2.793 to three decimal places.

因此梯形法则给出约 2.7931，即三位有效数字的 2.79，或三位小数的 2.793。

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7. Checking Against the Exact Value | 与精确值对照

Because √x a power of x, the integral can be evaluated exactly using the standard result ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) with n = ½.

由于 √x 是 x 的幂，可以用标准结果 ∫ xⁿ dx = xⁿ⁺¹ / (n + 1)（取 n = ½）求出精确值。

∫₁³ x^1/2 dx = [ ⅔ x^3/2 ]₁³ = ⅔ (3^3/2 − 1) = ⅔ (3√3 − 1)

Numerically, 3√3 so the exact value is ⅔ × 4.1961524 = 2.7974349, quoted as 2.7974 to four decimal places.

数值上，3√3 = 5.1961524，所以精确值为 ⅔ × 4.1961524 = 2.7974349，四位小数记作 2.7974。

Comparing the two results, the estimate 2.7930613 lies just below the exact value 2.7974349, which confirms the geometric prediction made from the concavity of the curve.

比较两个结果，估计值 2.7930613 略低于精确值 2.7974349，这验证了根据曲线凹性所作的几何预测。

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8. The Sign of the Error | 误差的正负

The error is defined here as the approximate value minus the exact value, so a negative error means the trapezium rule has underestimated the integral.

这里把误差定义为近似值减去精确值，因此负误差意味着梯形法则低估了积分值。

Error = 2.7930613 − 2.7974349 = −0.0043736

Percentage error = 0.0043736 / 2.7974349 × 100% ≈ 0.156%

answer, the reasoning matters more than the number. You should state that f″(x) = −1 / (4x^3/2 1, 3], that the curve is therefore concave down, and that each chord lies below the curve.

在考试作答中，推理过程比数字更重要。你应说明 f″(x) = −1 / (4x^3/2) 在 [1, 3] 上为负，因此曲线向下凹，每条弦都位于曲线下方。

an underestimate of the true area. If instead f″(x) were positive, as for y = x² on [0, 2

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