The Trapezium Rule | 梯形法则

📚 The Trapezium Rule | 梯形法则

Numerical integration allows us to approximate the value of a definite integral when an exact antiderivative is difficult or impossible to find. The trapezium rule is one of the most accessible methods, widely taught in Edexcel A‑Level Mathematics. By dividing the area under a curve into a series of trapezoids, it provides a reliable tool for estimating integrals using only sampled function values. Understanding its application, accuracy and limitations is essential for both Pure and Applied contexts.

数值积分使我们能够在难以找到精确原函数时近似计算定积分。梯形法则是Edexcel A‑Level数学中最常讲授的方法之一。它通过将曲线下方的区域分割为一系列梯形，仅利用离散函数值就能可靠地估算积分。理解其应用、精度和局限性对于纯数学和应用数学都至关重要。

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1. Introduction to Numerical Integration | 数值积分简介

In calculus, many definite integrals cannot be evaluated analytically. Functions such as e^−x², sin(x²) or √(1+x³) do not have elementary antiderivatives. When faced with such integrals, numerical methods bridge the gap. The idea is to replace the true area under y = f(x) between x = a and x = b with a sum of areas of simpler geometric shapes.

在微积分中，许多定积分无法解析求解。形如 e^−x²、sin(x²) 或 √(1+x³) 的函数没有初等原函数。面对这样的积分，数值方法弥补了这一不足。其思想是用简单几何图形面积之和来代替 y = f(x) 在 x = a 到 x = b 之间的真实面积。

The trapezium rule is a classic technique that uses straight-line segments to approximate the curve on each subinterval. It strikes a balance between simplicity and accuracy, making it a fundamental tool in any mathematician’s toolkit.

梯形法则是一种经典技术，在每个子区间上用直线段逼近曲线。它在简单性和准确性之间取得了平衡，是数学工具箱中的基本工具。

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

Compared with other numerical rules (e.g. rectangles or Simpson’s rule), the trapezium rule offers a straightforward formula that is easy to apply by hand or with a calculator. For Edexcel examinations, students are often required to set up the trapezium rule for a given function, choose an appropriate number of strips, and work with tabulated data. Its linear approximation on each segment gives an immediate geometric interpretation, which helps reinforce the concept of integration as a limit of sums.

与矩形法或辛普森法则相比，梯形法则公式简单，手工计算或使用计算器都很方便。在Edexcel考试中，学生经常需要为给定函数建立梯形法则、选择合适的条带数目，并处理表格数据。它在每个区间上的线性逼近具有直观的几何解释，有助于强化积分作为求和极限的概念。

Moreover, the trapezium rule sets the stage for understanding error behaviour. By comparing the trapezium estimate with the true integral, students gain insight into the effect of strip width and function curvature—skills that are valuable across both pure and applied modules.

此外，梯形法则为理解误差行为奠定了基础。通过比较梯形估计值与真实积分值，学生能深刻认识到条带宽度和函数曲率的影响——这些技能在纯数学和应用模块中都非常宝贵。

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

For a continuous function f(x) on [a, b], divide the interval into n equal strips of width h, where h = (b − a)/n. Let the x‑coordinates be x₀ = a, x₁ = a + h, …, xₙ = b, and let the corresponding function values be y₀ = f(x₀), y₁ = f(x₁), …, yₙ = f(xₙ). The trapezium rule approximates the integral as:

对于连续函数 f(x) 在区间 [a, b] 上，将该区间等分为 n 个宽度为 h 的小段，其中 h = (b − a)/n。设 x 坐标为 x₀ = a, x₁ = a + h, …, xₙ = b，对应的函数值为 y₀ = f(x₀), y₁ = f(x₁), …, yₙ = f(xₙ)。梯形法则将积分近似为：

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

This formula can be remembered as “half the strip width times (first + last + twice the sum of all the internal ordinates).” The factor of 2 comes from the fact that interior y‑values are shared by two adjacent trapezoids.

该公式可记忆为“条带宽度的一半乘以（首项 + 末项 + 两倍所有内部纵坐标之和）”。因子 2 的产生是因为内部 y 值被两个相邻梯形共享。

The expression is derived by summing the areas of n trapezoids, each with area ½h(yᵢ₋₁ + yᵢ). Collecting terms leads directly to the compact form above, which is extremely efficient for both algebraic and spreadsheet calculations.

该表达式由 n 个梯形的面积求和得出，每个梯形面积为 ½h(yᵢ₋₁ + yᵢ)。合并同类项后直接得到上述紧凑形式，对于代数计算和电子表格计算都十分高效。

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4. Understanding the Formula Derivation | 理解公式推导

Consider one strip between xᵢ₋₁ and xᵢ. The area of the single trapezoid is ½ × (width) × (sum of parallel sides) = ½h (yᵢ₋₁ + yᵢ). Summing over all n strips gives:

考虑 xᵢ₋₁ 和 xᵢ 之间的一个条带。单个梯形面积为 ½ × (宽度) × (上下底之和) = ½h (yᵢ₋₁ + yᵢ)。对所有 n 个条带求和得到：

Total ≈ ½h [(y₀ + y₁) + (y₁ + y₂) + … + (yₙ₋₁ + yₙ)]

Observe that y₁, y₂, …, yₙ₋₁ each appear twice, while y₀ and yₙ appear once. Factoring this out yields h/2 [y₀ + yₙ + 2(y₁ + … + yₙ₋₁)], which is precisely the trapezium rule. The derivation reinforces why symmetry in the formula is a natural consequence of the geometry.

注意到 y₁, y₂, …, yₙ₋₁ 各出现两次，而 y₀ 和 yₙ 只出现一次。提取公因子便得到 h/2 [y₀ + yₙ + 2(y₁ + … + yₙ₋₁)]，这正是梯形法则。该推导强调了公式的对称性为何是几何结构的自然结果。

Understanding this derivation is not just an algebraic exercise; it is the key to applying the rule correctly when ordinates are given directly in a table, or when only specific y‑values are known.

理解推导不仅仅是代数练习；当纵坐标直接以表格给出，或仅知道特定 y 值时，这是正确应用该法则的关键。

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5. Choosing the Number of Strips (n) | 选择条带数目

Increasing n makes h smaller and generally improves accuracy, because the straight‑line segments follow the curve more closely. However, a larger n also means more function evaluations, which can be time‑consuming. In Edexcel exam questions, n is often specified (e.g. “use 4 strips”), but you may also be asked to suggest a suitable number based on a desired accuracy or from a given table width.

增加 n 会使 h 变小，通常能提高准确性，因为直线段更贴近曲线。然而，更大的 n 也意味着更多的函数求值，可能较为耗时。在Edexcel考试题中，n 通常已指定（例如“使用4个条带”），但也可能要求根据期望精度或给定的表格宽度建议合适的数目。

When a table of values is provided, n is simply the number of intervals between the listed x‑values. Always check whether the x‑values are equally spaced; the trapezium rule can be adapted for uniform width only, or else a more general approach is needed.

当提供数值表格时，n 即为所列 x 值之间的区间数目。务必检查 x 值是否等距；梯形法则仅适用于均匀宽度，否则需要更一般的方法。

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6. Worked Example: Approximating a Definite Integral | 例题：近似定积分

Let us approximate ∫₀¹ √(1 + x³) dx using the trapezium rule with n = 4 strips.

让我们使用 n = 4 个条带的梯形法则来近似 ∫₀¹ √(1 + x³) dx。

First, h = (1 − 0)/4 = 0.25. The x‑ordinates and corresponding y‑values are recorded in a table:

首先，h = (1 − 0)/4 = 0.25。将 x 坐标及对应的 y 值记录在表格中：

xᵢ	0	0.25	0.5	0.75	1
yᵢ	1	1.0078	1.0607	1.1996	1.4142

The sum of the interior y‑values is 1.0078 + 1.0607 + 1.1996 = 3.2681. Applying the formula:

内部 y 值之和为 1.0078 + 1.0607 + 1.1996 = 3.2681。应用公式：

Approximation = (0.25/2) × [1 + 1.4142 + 2 × 3.2681]

= 0.125 × (2.4142 + 6.5362) = 0.125 × 8.9504 ≈ 1.1188

Therefore the trapezium rule gives I ≈ 1.1188 (to 4 decimal places). This process is straightforward and can be checked with a calculator for accuracy, but remember that the true value is around 1.111, illustrating a small overestimate due to the curve’s concavity.

因此梯形法则给出 I ≈ 1.1188（精确到4位小数）。这一过程直接明了，可用计算器检查准确性，但请记住真实值约为1.111，这表明由于曲线的凹性导致了轻微高估。

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7. Accuracy and Error Bounds | 精度与误差限

The error in the trapezium rule can be bounded using the second derivative of f. If f″ is continuous on [a, b] and |f″(x)| ≤ M for all x in the interval, then the absolute error satisfies:

梯形法则的误差可利用 f 的二阶导数进行界定。如果 f″ 在 [a, b] 上连续，且对于区间内的所有 x 有 |f″(x)| ≤ M，则绝对误差满足：

|Error| ≤ M(b − a)³ / (12n²)

This bound is not required for every Edexcel problem, but understanding it helps explain why doubling n reduces the error approximately by a factor of 4. The denominator 12n² means that error decreases quadratically with the number of strips—a powerful incentive to use a reasonable n.

这一误差限并非每道Edexcel题目都必须使用，但理解它有助于解释为何 n 加倍时误差大约减小为原来的四分之一。分母 12n² 意味着误差随条带数目平方递减——这是使用合理 n 值的强大动力。

In practice, exam questions may ask you to comment on accuracy by comparing with the exact value when available, or to recognise that increasing n improves the approximation. You might also be asked to find an upper bound for the error given M.

在实际考题中，可能会要求你将近似值与精确值（若已知）进行对比来评述精度，或认识到增加 n 会改善近似。也可能要求你在已知 M 时求出误差上限。

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8. Overestimates and Underestimates | 高估与低估

Whether the trapezium rule overestimates or underestimates the true area depends on the concavity of the function. If f″(x) > 0 (convex upwards), the trapezoids lie above the curve, causing an overestimate. If f″(x) < 0 (concave downwards), the trapezoids lie below the curve, resulting in an underestimate.

梯形法则是高估还是低估真实面积取决于函数的凹性。若 f″(x) > 0（向上凸），梯形位于曲线上方，导致高估。若 f″(x) < 0（向下凹），梯形位于曲线下方，导致低估。

This geometric insight is often assessed in examinations: given a graph or a sign of f″, you must deduce the nature of the trapezium estimate relative to the true integral. It provides a quick check without computing the exact integral.

这种几何见解在考试中常被考查：根据图形或 f″ 的符号，你必须推断梯形估计值相对于真实积分的性质。这提供了一个无需计算精确积分的快速检验方法。

For example, with f(x) = √(1 + x³), f″(x) > 0 on [0, 1], so the trapezium rule gives an overestimate, consistent with our worked example where 1.1188 > 1.111.

例如，对于 f(x) = √(1 + x³)，在 [0, 1] 上 f″(x) > 0，因此梯形法则给出高估值，这与我们例题的结果 1.1188 > 1.111 相符。

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9. Using the Trapezium Rule with Tables | 使用表格应用梯形法则

Many Edexcel questions provide a table of x and y values without explicitly stating the function. The trapezium rule can be applied directly. Suppose the table lists (x₀, y₀), (x₁, y₁), …, (xₙ, yₙ) with uniform spacing h. The estimated area is h times the average of the first and last y-values plus the sum of the interior y-values—or more formally, the structured formula h/2 [y₀ + yₙ + 2 Σᵢ₌₁ⁿ⁻¹ yᵢ].

许多Edexcel题目会给出 x 和 y 值的表格而未明确函数表达式。梯形法则可直接应用。假设表格列出了等距 h 的 (x₀, y₀), (x₁, y₁), …, (xₙ, yₙ)。估计面积等于 h 乘以首尾 y 值的平均与内部 y 值之和——或者更正式地使用结构化公式 h/2 [y₀ + yₙ + 2 Σᵢ₌₁ⁿ⁻¹ yᵢ]。

Students should be careful when reading the table: check whether the x‑values jump by equal steps. If the spacing is not uniform, the simple trapezium rule cannot be used without modification.

学生在阅读表格时应仔细检查：x 值是否等步长递增。若间距不统一，则不能不经修改就使用简单梯形法则。

A helpful tactic is to draw a quick sketch of the trapezoids to visualise the sum. This can prevent misapplication, especially when the table includes a mix of increasing and decreasing function values.

一个有用的策略是快速画出梯形示意图以直观显示求和过程。这能防止误用，特别是在表格包含递增与递减函数值混合时。

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10. Applications in Real-World Contexts | 实际应用

Beyond pure mathematics, the trapezium rule finds use in physics, engineering and economics whenever data is collected at discrete intervals. Calculating the area under a velocity‑time graph to find displacement, estimating total rainfall from a hydrograph, or determining work done from a force‑displacement curve all rely on numerical integration techniques akin to the trapezium rule.

除纯数学之外，每当数据以离散间隔采集时，梯形法则就应用于物理、工程和经济学中。计算速度‑时间图下方面积以求位移、通过水文过程线估算总降雨量，或由力‑位移曲线确定做功，都依赖于类似于梯形法则的数值积分技术。

In A‑Level Mechanics and Statistics, you may encounter contextual problems where the trapezium rule must be applied to tabulated data. Mastery of the method in Pure Mathematics therefore translates directly to improved modelling skills across the entire Edexcel specification.

在A‑Level力学和统计学中，你可能会遇到需将梯形法则应用于表格数据的背景问题。因此，纯数学中对该方法的掌握能直接转化为整个Edexcel考纲中建模能力的提升。

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11. Limitations and Alternatives | 局限性与替代方法

The trapezium rule’s principal limitation is its linear approximation; for highly oscillatory or rapidly changing functions, many strips may be needed to achieve acceptable accuracy, and even then the error can be significant. In such cases, more sophisticated methods like Simpson’s rule (which uses parabolic arcs) are more efficient, though not required in the same depth at A‑Level.

梯形法则的主要局限在于其线性逼近；对于高度振荡或变化剧烈的函数，可能需要许多条带才能达到可接受的精度，且即便如此误差也可能很大。此时，更高级的方法如辛普森法则（使用抛物线弧段）更为高效，尽管 A‑Level 对其深度不作同样要求。

Another limitation is the requirement of equally spaced abscissae. If data is irregularly spaced, the trapezium rule must be generalised, or another technique such as the weighted trapezium rule employed. Students are advised to recognise the constraints and to appreciate when a numerical result should be treated with caution.

另一局限是要求等距横坐标。若数据间距不规则，则必须推广梯形法则，或采用诸如加权梯形法则等其他技术。建议学生认识到这些约束，并懂得在哪些情况下应对数值结果持谨慎态度。

Alternatives like the midpoint rule or Gaussian quadrature are beyond the A‑Level syllabus, but knowing that the trapezium rule is part of a broader family of Newton–Cotes formulas can broaden a student’s perspective.

虽然中点法则或高斯求积等替代方法超出 A‑Level 大纲，但知道梯形法则是更广泛的牛顿‑柯特斯公式家族的一部分，有助于学生开阔视野。

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12. Exam Tips for Edexcel A-Level | Edexcel A-Level 考试技巧

In Edexcel exams, the trapezium rule commonly appears in Pure Mathematics Paper 1 or Paper 2. Key tips include: (1) always write down h and n clearly; (2) show a table and label yᵢ values; (3) write the formula before substituting numbers to secure method marks; (4) round final answers only at the end to avoid cumulative rounding errors; (5) when asked to “comment on accuracy”, relate your answer to the concavity or to the effect of increasing strips.

在Edexcel考试中，梯形法则通常出现在纯数学试卷1或试卷2中。关键技巧包括：（1）务必清晰地写出 h 和 n；（2）展示表格并标注 yᵢ 值；（3）代入数值前先写出公式，以确保方法分；（4）仅在最后一步对最终答案四舍五入，避免累积舍入误差；（5）当被要求“评述精度”时，将答案与凹性或增加条带的效果联系起来。

Additionally, if a question states “use the trapezium rule with the values given in the table”, ensure you identify the correct h and count the number of strips correctly. A common mistake is confusing n (number of strips) with the number of ordinates (n + 1). Keep your working systematic and underline the final approximate integral value.

此外，若题目说明“利用表格给出的值使用梯形法则”，请确保正确识别 h 并准确计数条带数目。常见错误是将 n（条带数）与纵坐标数目（n + 1）混淆。保持解题步骤条理清晰，并在最终近似积分值下画线。

Finally, time management is important. Solving a trapezium rule question usually takes 5–8 minutes; practicing with past papers will build speed and confidence.

最后，时间管理很重要。解答梯形法则题目通常需要 5–8 分钟；通过历年真题练习可提升速度与信心。

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