📚 The Trapezium Rule for A-Level Maths | A-Level数学梯形法则
The trapezium rule is a fundamental numerical integration technique in A-Level Mathematics. It provides a way to estimate the area under a curve when the function cannot be integrated analytically or when only a set of data points is available.
梯形法则是A-Level数学中的基本数值积分方法。当函数无法解析积分或只有一组数据点时，它可以估算曲线下的面积。
1. Introduction to the Trapezium Rule | 梯形法则简介
Named after the trapeziums (trapezoids) used to approximate the area, the rule divides the area into a number of vertical strips. Each strip is treated as a trapezium, and the sum of their areas gives the estimate.
该方法因使用梯形来近似面积而得名。它将区域分成若干垂直条带，每个条带看作一个梯形，其面积之和即为估计值。
The approximation improves as the number of strips increases. At A-Level, you are typically asked to use 4 or 5 ordinates to keep calculations manageable.
随着条带数量增加，近似值会更精确。在A-Level考试中，通常要求使用4或5个坐标点，以便计算。
2. Derivation of the Formula | 公式推导
trips are x₀ = a, x₁ = a + h, ..., xₙ = b. The corresponding y-values are y₀, y₁, ..., yₙ.
考虑曲线 y = f(x) 从 x = a 到 x = b 下方的面积。将区间分成 n 个等宽条带，宽度 h = (b - a)/n。条带的端点为 x₀ = a, x₁ = a + h, ..., xₙ = b。相应的 y 值为 y₀, y₁, ..., yₙ。
The area of a trapezium with parallel sides yₖ₋₁ and yₖ and width h is h/2 (yₖ₋₁ + yₖ). Summing over all strips gives the trapezium rule.
上、下底为 yₖ₋₁ 和 yₖ，宽为 h 的梯形面积为 h/2 (yₖ₋₁ + yₖ)。对所有条带求和即得梯形法则。
∫ₐᵇ f(x) dx ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]
Notice that the first and last ordinates are multiplied by 1, while the interior ordinates are multiplied by 2. This pattern is the essence of the rule.
请注意，第一个和最后一个坐标乘以 1，中间坐标乘以 2。这一模式是法则的核心。
3. Understanding Strip Width | 理解条带宽度
width h is calculated as h = (b - a) / n. Always check that the interval is divided evenly – umber of strips instead of ordinates. If you have n strips, you will have n + 1 ordinates.
条带宽度 h 计算为 h = (b - a) / n。务必确认区间是等分的——常见的错误是用梯形数量代替坐标点数量。如果有 n 个梯形，就会有 n + 1 个坐标。
In exam questions, the number of strips is often stated directly, or you may be asked to use a given set of x-values from a table. Make sure you can identify h even if the intervals are not uniform (though the standard rule assumes equal width).
在考试题目中，梯形数量通常直接给出，或者要求你用表格中给定的一组 x 值。即使区间不完全均匀（标准法则假设等宽），你也应能识别出 h。
4. Worked Example: Approximating a Simple Integral | 例题：近似简单积分
We will estimate I = ∫₀² √1 + x³ dx using the trapezium rule with 4 strips. The interval width is h = (2 - 0)/4 = 0.5.
我们将使用梯形法则以 4 条带估算 I = ∫₀² √1 + x³ dx。区间宽度 h = (2 - 0)/4 = 0.5。
We need the y-values corresponding to x = 0, 0.5, 1, 1.5, 2. Calculate each:
y(0) = √1 = 1
y(0.5) = √1 + 0.125 = √1.125 ≈ 1.06066
y(1) = √2 ≈ 1.41421
y(1.5) = √1 + 3.375 = √4.375 ≈ 2.09165
y(2) = √1 + 8 = √9 = 3
我们需要 x = 0, 0.5, 1, 1.5, 2 对应的 y 值。计算如下：
y(0) = √1 = 1
y(0.5) = √1 + 0.125 = √1.125 ≈ 1.06066
y(1) = √2 ≈ 1.41421
y(1.5) = √1 + 3.375 = √4.375 ≈ 2.09165
y(2) = √1 + 8 = √9 = 3
| x | 0 | 0.5 | 1 | 1.5 | 2 |
|---|---|---|---|---|---|
| y | 1 | 1.06066 | 1.41421 | 2.09165 | 3 |
65) + 3] = 0.25 [1 + 2(4.56652) + 3] = 0.25 [1 + 9.13304 + 3] = 0.25 × 13.13304 = 3.28326.
应用梯形法则：I ≈ 0.25 × [1 + 2×(1.06066 + 1.41421 + 2.09165) + 3] = 0.25 × 13.13304 = 3.28326。
A more accurate value obtained by numerical integration is approximately 3.241. The trapezium rule gives an overestimate here because the curve is concave up.
通过数值积分获得的更精确值约为 3.241。这里梯形法则高估了，因为曲线是凹向上的。
5. Using a Table of Values | 使用数值表
In the exam, always present your workings in a clear table. List x, the corresponding y-values, and then apply the weighting coefficients (1 for ends, 2 for interior). This reduces arithmetic errors.
考试中务必用清晰表格展示步骤。列出 x、对应的 y 值，然后应用权重系数（两端乘 1，中间乘 2）。这可减少算术错误。
ient and a row for the product coefficient × y. Then sum the products and multiply by h/2.
例如，在上表中增加一行系数和一行系数乘 y 的积，然后求和并乘以 h/2。
This method is highly recommended by examiners and makes your method easy to follow.
阅卷老师强烈推荐这种方法，能让你的解题过程一目了然。
6. Increasing Accuracy with More Strips | 增加梯形数量以提升精度
mate; doubling to n = 8 would reduce the error further, though the computation load grows.
随着 n 增加，近似值通常会改善。n = 4 已经得到了合理的估计；加倍到 n = 8 会进一步减小误差，但计算量也会增加。
In Edexcel A-Level exams, questions sometimes ask you to compare results for different strip numbers or to explain how the accuracy changes. You should note that the trapezium rule tends to overestimate for convex functions and underestimate for concave functions if the curve lies above the axis.
在 Edexcel A-Level 考试中，有时会让你比较不同梯形数的结果或解释精度如何变化。应注意，若曲线位于 x 轴上方，梯形法则对于凸函数倾向于高估，对于凹函数倾向于低估。
7. Error Estimation and Bounds | 误差估计与界限
Although A-level does not require a rigorous error formula, understanding the sign of the error can gain marks. If the second derivative f''(x) > 0 (convex), the trapezoidal approximation generally overestimates; if f''(x) < 0 (concave), it underestimates.
虽然A-Level不要求严格的误差公式，理解误差的符号仍可得分。如果二阶导数 f''(x) > 0（凸函数），梯形近似通常高估；若 f''(x) < 0（凹函数），则低估。
In our example, f(x) = √1+x³ is convex on [0,2], explaining the overestimate. You can mention this in an 'explain' type question.
在我们的例子中，f(x) = √1+x³ 在 [0,2] 上是凸函数，这解释了高估现象。你可以在“解释”类型的题目中提到这点。
8. Applying the Rule to Curves That Dip Below the x-axis | 曲线部分在x轴下方的应用
The trapezium rule calculates the signed area, so parts of the curve below the x-axis will produce negative y-values. These are included in the sum with their signs, giving the net area.
梯形法则计算的是带符号的面积，所以曲线在 x 轴下方的部分会产生负的 y 值。这些值带着符号计入求和，得到净面积。
If the question asks for the total actual area, you must break the interval at the roots and work out each part separately, taking absolute values of the areas.
若题目要求实际总面积，必须在根处分段，分别计算各部分面积，并取绝对值相加。
9. Exam-Style Problem | 考试风格题目
apezium rule with 4 strips to estimate ∫₁³ ln(x) dx. Give your answer to 4 decimal places.
题目：使用 4 条带的梯形法则估算 ∫₁³ ln(x) dx，答案保留四位小数。
Solution: a = 1, b = 3, n = 4, so h = (3 - 1)/4 = 0.5.
x-values: 1, 1.5, 2, 2.5, 3.
y = ln(x): ln(1)=0, ln(1.5)≈0.405465, ln(2)≈0.693147, ln(2.5)≈0.916291, ln(3)≈1.098612.
解答：a = 1, b = 3, n = 4，故 h = 0.5。x 值：1, 1.5, 2, 2.5, 3。y = ln(x) 值：0, 0.405465, 0.693147, 0.916291, 1.098612。
(2.014903) + 1.098612] = 0.25 × (4.029806 + 1.098612) = 0.25 × 5.128418 = 1.2821 (4 d.p.).
应用法则：I ≈ 0.25 × [0 + 2×(0.405465+0.693147+0.916291) + 1.098612] = 0.25 × 5.128418 = 1.2821。（四位小数）
58. The trapezium estimate is slightly lower, consistent with the concave nature of ln(x).
精确值为 3ln3 - 2 ≈ 1.2958。梯形估计值略低，这与 ln(x) 凹函数性质一致。
10. Common Mistakes to Avoid | 常见错误及避免方法
One of the most frequent errors is forgetting to divide by 2 when applying the formula. Always write h/2 explicitly.
最常见的错误之一是在应用公式时忘记除以 2。务必明确写出 h/2。
Another mistake is miscounting the number of strips. If the question says 'use 4 strips', you must have 5 ordinates. Confusing these leads to a wrong h or missing terms.
另一个错误是数错梯形数量。如果题目说“用 4 条带”，必须有 5 个坐标点。混淆了会导致 h 错误或缺少项。
Also, avoid rounding intermediate y-values too early – keep sufficient decimals to maintain accuracy in the final estimate.
此外，避免过早舍入中间的 y 值——保留足够小数以保证最终估计的精度。
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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