📚 The Trapezium Rule | 梯形法则
In A-Level Mathematics, not every definite integral can be evaluated using standard functions or the integration techniques in the specification. The trapezium rule provides a numerical method for approximating the area under a curve by dividing the region into trapeziums of equal width.
在 A-Level 数学中，并非所有定积分都能用标准函数或大纲中的积分方法求值。梯形法则是一种数值方法，通过将曲线下方区域划分为等宽梯形来近似曲线下的面积。
1. What is the trapezium rule? | 什么是梯形法则？
The trapezium rule approximates the definite integral ∫ₐᵇ f(x) dx by replacing the region under the curve y = f(x) with a series of trapeziums of equal width. Each trapezium uses two adjacent points on the curve as its top corners, and the straight line between them forms the top edge.
梯形法则通过将曲线 y = f(x) 下方的区域替换为一组等宽梯形来近似定积分 ∫ₐᵇ f(x) dx。每个梯形以曲线上两个相邻点作为上方的角，两点之间的直线构成梯形的上边。
The area of one trapezium of width h with parallel side lengths yᵢ₋₁ and yᵢ is h/2 (yᵢ₋₁ + yᵢ). Adding these individual areas gives the trapezium rule estimate for the whole interval.
宽度为 h、平行边长为 yᵢ₋₁ 和 yᵢ 的单个梯形面积为 h/2 (yᵢ₋₁ + yᵢ)。将这些单个面积相加，就得到整个区间上的梯形法则估计值。
2. When is it used? | 何时使用梯形法则？
You should use the trapezium rule when the function cannot be integrated analytically using the techniques in the A-Level syllabus. Common examples include functions such as e^(x²), where no elementary antiderivative exists, or functions defined only by a table of measured values rather than an algebraic expression.
当函数无法用 A-Level 大纲中的方法进行解析积分时，应使用梯形法则。常见例子包括 e^(x²) 等不存在初等原函数的函数，或者仅由测量数据表给出而没有代数表达式的函数。
It is also useful in modelling questions where a curve is approximated from discrete readings, and it can be used to check whether an exact answer produced by another method is reasonable.
在建模题中，如果曲线由离散读数近似得到，梯形法则也很有用。它还可以用来检验其他方法得到的精确答案是否合理。
3. The formula and notation | 公式与符号
Suppose the interval [a, b] is divided into n equal strips of width h = (b − a) / n. Write x₀ = a, x₁ = a + h, …, xₙ = b and yᵢ = f(xᵢ). The trapezium rule estimate is
设区间 [a, b] 被分成 n 个等宽小区间，宽度 h = (b − a) / n。记 x₀ = a，x₁ = a + h，…，xₙ = b，且 yᵢ = f(xᵢ)。梯形法则的估计值为
∫ₐᵇ f(x) dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]
The same expression can be written in expanded form as
同一个表达式可以展开写为
∫ₐᵇ f(x) dx ≈ h/2 [y₀ + 2y₁ + 2y₂ + … + 2yₙ₋₁ + yₙ]
The first and last ordinates are multiplied by 1, and all intermediate ordinates are multiplied by 2 because they are shared by two adjacent trapeziums.
第一个和最后一个纵坐标的系数为 1，所有中间纵坐标的系数为 2，因为它们被两个相邻的梯形共用。
4. Setting up ordinates | 设置纵坐标
Before substituting values, identify whether the question gives the number of strips or the number of ordinates. If the question says “use four strips”, then n = 4, so h = (b − a) / 4 and you need the five x-values from x₀ to x₄.
在代入数值之前，先弄清题目给出的是区间数还是纵坐标数。如果题目说“使用四个区间”，则 n = 4，所以 h = (b − a) / 4，并且你需要 x₀ 到 x₄ 共五个 x 值。
If the question says “use five ordinates”, then there are five y-values and therefore four strips. In this case n = 4 and h = (b − a) / 4. This distinction is a very common source of errors.
如果题目说“使用五个纵坐标”，则有五个 y 值，因此有四个区间。此时 n = 4，h = (b − a) / 4。这一区别是常见的错误来源。
Always write the x-values in increasing order and calculate the corresponding y-values to at least four decimal places if using a calculator. Presenting a clear table is an effective way to organise the working.
应始终按递增顺序写出 x 值，并在使用计算器时至少保留四位小数计算相应的 y 值。清晰列出表格是整理计算过程的有效方法。
5. Worked example: four strips | 例题：四个区间
Use the trapezium rule with four strips to estimate ∫₁³ 1/x dx.
用四个区间下的梯形法则估计 ∫₁³ 1/x dx。
Here a = 1, b = 3, n = 4, so h = (3 − 1) / 4 = 0.5. The ordinates are x = 1, 1.5, 2, 2.5, 3. The corresponding y-values are shown in the table.
这里 a = 1，b = 3，n = 4，所以 h = (3 − 1) / 4 = 0.5。纵坐标为 x = 1、1.5、2、2.5、3。对应的 y 值见下表。
| x | 1 | 1.5 | 2 | 2.5 | 3 |
|---|---|---|---|---|---|
| y = 1/x | 1 | 0.6667 | 0.5 | 0.4 | 更多咨询请联系16621398022（同微信）
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