📚 The Trapezium Rule | 梯形法则
When a definite integral cannot be evaluated analytically or when a function is only known at certain points, numerical methods provide a way to approximate the area under a curve. The trapezium rule is one of the simplest and most widely used techniques for estimating definite integrals. By dividing the area into a series of trapezoids, we can sum their areas to obtain an approximation that becomes more accurate as the number of subdivisions increases.
当定积分无法通过解析方法求出，或者函数只在某些离散点已知时，数值方法为逼近曲线下的面积提供了途径。梯形法则是最简单、最常用的数值积分方法之一。通过将区域分割成若干梯形并求其面积之和，可以得到积分的近似值，且分割越细近似越精确。
1. Introduction to Numerical Integration | 数值积分导论
(x²) dx or functions given as data tables. The trapezium rule provides a straightforward remedy: split the interval into equal subintervals, construct trapezia using the function values at the endpoints, and sum their areas. Unlike exact integration, the result is an estimate, but the method gives a clear geometric interpretation.
在IB数学中，学生经常会遇到不能用初等函数表达的定积分，例如 ∫ sin(x²) dx 或由数据表给出的函数。梯形法则提供了一种简单的处理方式：将积分区间等分为若干子区间，利用端点处的函数值构建梯形，再将它们的面积相加。与精确积分不同，结果是一个近似值，但该方法有着清晰的几何解释。
2. The Trapezium Rule Formula | 梯形法则公式
equal subintervals, each of width h = (b − a)/n. Let the x-coordinates be x₀ = a, x₁, x₂, …, xₙ = b, and let y₀ = f(x₀), y₁ = f(x₁), … , yₙ = f(xₙ). The trapezium rule states:
假设我们要近似 ∫ₐᵇ f(x) dx。将区间 [a, b] 分成 n 个等宽子区间，宽度 h = (b − a)/n。记 x 坐标为 x₀ = a, x₁, x₂, …, xₙ = b，函数值为 y₀, y₁, …, yₙ。梯形法则的公式为：
∫ₐᵇ f(x) dx ≈ ½ h [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]
The coefficient of the first and last ordinates is 1; all interior ordinates appear with a coefficient 2. This weighted pattern is central to the formula and appears in every IB examination question on the topic.
首尾两个纵坐标的系数为 1，所有内部纵坐标的系数为 2。这种加权模式是公式的核心，并出现在每一道涉及该专题的IB考题中。
3. Derivation of the Trapezium Rule | 梯形法则的推导
trip is ½ (yᵢ₋₁ + yᵢ) × h. Summing over all n strips gives ½ h (y₀ + y₁) + ½ h (y₁ + y₂) + … + ½ h (yₙ₋₁ + yₙ). Simplifying, y₁, y₂, …, yₙ₋₁ each appear twice, while y₀ and yₙ appear once, yielding the compact form above.
每个子区间 [xᵢ₋₁, xᵢ] 形成一个宽度为 h 的窄条。该条上的梯形面积为 ½ (yᵢ₋₁ + yᵢ) × h。对所有 n 个窄条求和得 ½ h (y₀ + y₁) + ½ h (y₁ + y₂) + … + ½ h (yₙ₋₁ + yₙ)。整理后，y₁, y₂, …, yₙ₋₁ 各出现两次，而 y₀ 和 yₙ 只出现一次，从而得到上述紧凑形式。
This derivation is particularly helpful when the interval is not divided equally; the standard formula still applies when all strips share the same width h. If the function values come from an unequal grid, the composite trapezium rule can be adapted by using each subinterval's own h, but IB questions almost always work with equal h.
这一推导在区间不等宽时尤为有用；当所有窄条宽度相等时，标准公式即可直接使用。如果函数值来自非等距网格，可以通过各自子区间的 h 对复合梯形法则进行修正，但IB题目几乎总是处理等距h的情况。
4. Applying the Trapezium Rule: Step-by-Step | 应用步骤
Follow these steps when tackling a trapezium rule problem:
解答梯形法则问题时请遵循以下步骤：
- Identify a, b and n: check the limits of integration or the number of data points provided. Often n is the number of strips, not the number of ordinates.
- 找到 a, b 和 n：确认积分上下限或给出的数据点数目。注意 n 是窄条数目，而非纵坐标数目。
- Calculate h: − a)/n. If a table of values is given, h is simply the constant spacing between x-values.
- 计算 h：h = (b − a)/n。若给出数值表，则 h 即为 x 值之间的固定间距。
- List the y-values: y₀, y₁, y₂, …, yₙ.
- 列出 y 值：y₀, y₁, y₂, …, yₙ。
- Apply the formula: compute ½ h [first + last + 2 × (sum of interior ordinates)].
- 套用公式：计算 ½ h [首项 + 末项 + 2 × (内部纵坐标之和)]。
- Interpret the result: state the approximation and mention whether it is an overestimate or underestimate if the function's concavity is known.
- 解读结果：指出近似值，并在已知凹凸性的情况下说明是高估还是低估。
5. Worked Example with Equal Intervals | 等距区间例题
Example: mate ∫₁⁵ 1/x dx using the trapezium rule with 4 strips. Compare with the exact value ln 5.
例题：用梯形法则，取 4 个窄条，近似 ∫₁⁵ 1/x dx，并与精确值 ln 5 比较。
5, n = 4, so h = (5 − 1)/4 = 1. x-values: 1, 2, 3, 4, 5. y-values: 1, 1/2, 1/3, 1/4, 1/5.
此处 a = 1, b = 5, n = 4，故 h = (5 − 1)/4 = 1。x 值：1, 2, 3, 4, 5。y 值：1, 1/2, 1/3, 1/4, 1/5。
+ 1/5 + 2 × (13/12)] = 0.5 × [1.2 + 26/12] = 0.5 × [1.2 + 2.1667] = 0.5 × 3.3667 ≈ 1.6833.
内部纵坐标之和 = 1/2 + 1/3 + 1/4 = 13/12。总近似值 + 1/5 + 2 × (13/12)] = 0.5 × [1.2 + 26/12] = 0.5 × [1.2 + 2.1667] = 0.5 × 3.3667 ≈ 1.6833。
ezium rule with 4 strips overestimates because the curve y = 1/x is concave up on [1,5] (f''(x) > 0).
精确值为 ln 5 ≈ 1.6094。用 4 个窄条的梯形法则高估了真实值，因为曲线 y = 1/x 在 [1,5] 上是凹向上的（f''(x) > 0）。
6. Using the Trapezium Rule with a Table of Values | 使用数值表进行梯形近似
IB examination papers frequently provide a table of x and y values instead of an explicit function. In such cases, confirm that the x-values are equally spaced. If the spacing is constant, the same formula applies directly; y₀ is the first y‑value, yₙ is the last.
IB考卷常常会给出 x 与 y 的数值表而非显式函数。此时需要确认 x 值是否等距。若间距恒定，则可直接套用同一公式；y₀ 为第一个 y 值，yₙ 为最后一个。
| x | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 |
| y | 2.1 | 3.6 | 5.3 | 7.2 | 9.1 |
5. Always state the calculated figure and, if asked, deduce whether it over- or underestimates the true area.
取 h = 0.5，估计面积为 ½ × 0.5 × [2.1 + 9.1 + 2 × (3.6+5.3+7.2)] = 0.25 × [11.2 + 2×16.1] = 0.25 × 43.4 = 10.85。务必给出计算值，并在题目要求时判断其是对真实面积的高估还是低估。
7. Overestimates and Underestimates – The Role of Concavity | 高估与低估——凹凸性的作用
Whether the trapezium rule overestimates or underestimates the true integral depends on the function's curvature. If the function is concave up (convex, f''(x) > 0) on the interval, the top edges of the trapezia lie above the curve, so the rule gives an overestimate. If the function is concave down (f''(x) < 0), the trapezia lie below the curve, producing an underestimate.
梯形法则是高估还是低估真实积分取决于函数的弯曲方向。若函数在区间上是凹向上（凸，f''(x) > 0），梯形的上底位于曲线上方，故结果为高估。若函数凹向下（f''(x) < 0），梯形位于曲线下方，结果为低估。
IB questions often ask students to sketch the trapezia on a given graph and state whether the approximation is too large or too small. Memorising the sign of the second derivative for common functions (e.g. 1/x on x > 0 is concave up) is a useful shortcut.
IB考题常要求学生绘制梯形在给定函数图像上的位置，并指出近似值是偏大还是偏小。熟记常见函数二阶导数的正负（如 x > 0 时 1/x 为凹向上）是一个有效的解题技巧。
8. Error Bounds for the Trapezium Rule | 梯形法则的误差界
The error in the trapezium rule can be bounded if the second derivative of f is continuous. The absolute error E satisfies:
若 f 的二阶导数连续，梯形法则的误差可以给出一个上界。绝对误差 E 满足：
|E| ≤ (b − a)³ / (12n²) × max{ |f''(x)| : a ≤ x ≤ b }
This formula shows that doubling the number of strips reduces the error bound by roughly a factor of 4, since n² appears in the denominator. While IB examinations do not require an algebraic proof of this bound, it explains why using more strips dramatically improves precision.
由该公式可见，窄条数目加倍，可使误差界大约缩小为原来的四分之一，因为分母中含有 n²。虽然IB考试不要求对此误差界进行代数证明，但它解释了为何增加窄条数量会显著提高精度。
9. Increasing the Number of Strips – Accuracy and Convergence | 增加窄条数量——精度与收敛性
you might be asked to repeat a calculation with twice the number of strips and observe the improvement.
随着 n 增加，h 减小，梯形拼接的形状越来越贴近曲线。对于连续函数，当 n → ∞ 时梯形法则的近似值收敛于真实积分。在实际IB问题中，可能会要求用两倍数量的窄条重新计算，并观察精度的提升。
the estimate to approximately 1.617. Such sequences can help students appreciate the concept of convergence without formal limits.
在前述 ∫₁⁵ 1/x dx 的例子中，取 n = 8 时 h = 0.5，估计值约为 1.632，更贴近 ln 5 ≈ 1.609。再次加倍 n，估计值接近 1.617。这类序列能帮助学生直观体会收敛的概念，而无需进行形式化的极限运算。
10. Comparison with Simpson’s Rule | 与辛普森法则的简要对比
Simpson's rule uses parabolic arcs instead of straight lines to approximate the curve, and it often yields higher accuracy with fewer intervals. The trapezium rule is simpler, requiring only linear interpolation, and serves as a foundation for understanding more advanced quadrature methods. In the IB syllabus, the trapezium rule may appear alongside Simpson's rule, but they are assessed as separate techniques.
辛普森法则用抛物线弧而非直线来拟合曲线，通常用更少的区间就能达到更高精度。梯形法则更简单，只需线性插值，是理解更高级求积方法的基础。在IB课程中，梯形法则可能与辛普森法则一同出现，但二者会作为不同技巧来考查。
The order of the error is another distinction: trapezium error is O(1/n²), while Simpson's is O(1/n⁴) for smooth functions. When questions ask which method is likely more accurate given the same number of strips, the answer is generally Simpson's rule.
二者的误差阶也不同：梯形法则误差为 O(1/n²)，而辛普森法则对光滑函数为 O(1/n⁴)。当试题询问在相同窄条数下哪种方法可能更精确时，答案通常为辛普森法则。
11. Typical IB Exam Question Patterns | IB考试常见题型
IB questions on the trapezium rule typically follow predictable patterns:
IB关于梯形法则的考题通常遵循几类固定模式：
- Direct calculation: Given a function and n, find the approximate integral.
- 直接计算：给出函数和 n，求近似积分。
- Table completion: Fill missing y-values and then apply the rule.
- 表格补全：填充缺失的 y 值，然后应用公式。
- Graphical interpretation: On a provided graph, draw the trapezia and state the nature of the approximation.
- 图形解读：在给定坐标图中画出梯形，并说明近似值的性质。
- Error analysis: Compare estimates for different n and comment on convergence.
- 误差分析：比较不同 n 下的估计值，并对收敛性作出评价。
- Combination with other topics: E.g., kinematics tables to estimate total distance, or using the result to bound an integral.
- 与其他专题结合：如利用运动学数据表估算总路程，或用近似结果对积分进行放缩。
Practising these patterns ensures both speed and accuracy in the examination.
熟练这些题型能确保考试中答题快速且准确。
12. Summary and Final Tips | 总结与备考建议
The trapezium rule is an essential numerical tool in IB Mathematics. Always identify h correctly, write down the formula before substituting numbers, and double-check whether you have used the coefficients 1, 2, 2, …, 1. When a question mentions concavity, link it to over‑ or underestimation. For error‑bound questions, remember the expression involves (b−a)³/(12n²) and the maximum second derivative magnitude. Finally, practise with both analytic functions and tables, as both appear regularly.
梯形法则是IB数学中的一项基本数值工具。务必准确辨认 h 的值，先写出公式再代入数字，并仔细检查是否使用了 1, 2, 2, …, 1 的系数。遇到凹凸性提示时，要将其与高估或低估联系起来。对于误差界问题，记住表达式含有 (b−a)³/(12n²) 和二阶导数的最大绝对值。最后，对函数式与数值表两种题型都要勤加练习，因为二者在考试中都很常见。
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