16C Trapezoidal Rule | 梯形法则

📚 16C Trapezoidal Rule | 梯形法则

In IB Mathematics, evaluating definite integrals analytically is not always possible. Many functions do not have elementary antiderivatives, or we may only know their values at discrete points. The trapezoidal rule is a fundamental numerical technique that approximates the area under a curve by dividing it into trapezoids. It is widely used in both Analysis & Approaches and Applications & Interpretation syllabuses, often appearing alongside problems involving data tables, velocity-time graphs, and real-world modelling. Mastering this rule strengthens your understanding of integration and prepares you for calculator-based and theoretical exam questions.

在 IB 数学中，并非所有定积分都能通过解析方法求出。许多函数没有初等原函数，或者我们只能获得函数在离散点处的值。梯形法则是一种基础的数值方法，通过将曲线下方的区域划分为梯形来逼近积分值。它在分析与方法和应用与解释课程大纲中都有广泛应用，常与数据表格、速度-时间图像和现实建模问题一同出现。掌握该法则可以加深对积分的理解，并为计算器操作和理论考试题目做好准备。

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1. What is Numerical Integration? | 什么是数值积分？

dx. It is essential when the function f has no closed-form antiderivative, such as f(x)=eˣ tion methods. It replaces the curve with straight-line segments, forming trapezoids whose combined area approximates the true integral.

数值积分是指一系列用于估算定积分 ∫ₐᵇ f(x) dx 的算法。当被积函数没有封闭形式的原函数时（例如 f(x)=eˣ² 或 f(x)=sin(x)/x），或者函数仅由实验数据给出时，数值积分就变得必不可少。梯形法则是其中最简单、最直观的方法之一。它用直线段代替曲线，形成一系列梯形，这些梯形的总面积逼近真实的积分值。

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2. Geometric Interpretation | 梯形法则的几何解释

In the single-interval trapezoidal rule, we draw a chord from (a,f(a)) to (b,f(b)) and calculate the area of the resulting trapezoid. This area equals the width (b−a) multiplied by the average height (f(a)+f(b))/2. Geometrically, we are approximating the curve by a straight line, which is reasonable over short intervals but can introduce larger errors when the function curves significantly.

想象一下 y=f(x) 图像与 x 轴在 x=a 到 x=b 之间围成的区域。在单区间梯形法则中，我们从 (a,f(a)) 到 (b,f(b)) 连一条弦，并计算所得梯形的面积。该面积等于宽度 (b−a) 乘以平均高度 (f(a)+f(b))/2。从几何上看，我们用一条直线来近似曲线，这对于短区间是合理的，但当函数弯曲程度较大时，可能会引入较大的误差。

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3. Deriving the Single-Interval Formula | 推导单一区间梯形公式

For one subinterval of width h=b−a, the area of the trapezoid is simply (h/2)⋅ rea of a trapezoid: base times average of the two parallel sides. In integration notation,

∫ₐᵇ f(x) dx ≈ (b−a)/2 [f(a)+f(b)]

This is the simplest form of the trapezoidal rule. It exactly matches the integral of the linear Lagrange interpolating polynomial through the endpoints. Although crude for large intervals, it forms the basis for the composite rule.

对于宽度为 h=b−a 的单个子区间，梯形面积就是 (h/2)⋅[f(a)+f(b)]。该公式直接源于梯形面积公式：底乘以两平行边的平均值。用积分符号表示为：

∫ₐᵇ f(x) dx ≈ (b−a)/2 [f(a)+f(b)]

这是梯形法则的最简形式，它精确等于通过两个端点的线性拉格朗日插值多项式的积分。尽管对于大区间显得粗糙，但它构成了复化法则的基础。

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4. The Composite Trapezoidal Rule | 复化梯形法则

.,n. Applying the single-interval formula to each subinterval and summing the areas gives:

∫ₐᵇ f(x) dx ≈ h/2 [f(x₀) + 2f(x₁) + 2f(x₂) + ⋯ + 2f(xₙ₋₁) + f(xₙ)]

Interior points are multiplied by 2 because they appear as the right endpoint of one strip and the left endpoint of the next. The first and last function values are multiplied by 1. This weighted sum is easy to program on a GDC or compute with a table of values.

为了提高精度，将区间 [a,b] 分成 n 个等宽子区间。令 h = (b−a)/n，且 xᵢ = a + i h（i=0,1,...,n）。对每个子区间应用单区间公式并累加面积，便得到：

∫ₐᵇ f(x) dx ≈ h/2 [f(x₀) + 2f(x₁) + 2f(x₂) + ⋯ + 2f(xₙ₋₁) + f(xₙ)]

内部点的函数值乘以 2，因为它们同时作为一个竖条的右端点和下一个竖条的左端点出现。首尾函数值则乘以 1。这种加权和在图形计算器上很容易编程，也便于用数据表计算。

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5. Step Size and Notation | 步长与符号

al role: smaller h generally yields a better approximation. In IB problems, n is often given directly (e.g., “use 4 strips” e x-values are equally spaced. Always check that x-values are equally spaced before applying the composite rule. Common notation includes x₀, x₁, …, xₙ for the nodes and y₀, y₁, …, yₙ for the corresponding function values or data points.

步长 h = (b−a)/n 起着核心作用：h 越小通常近似效果越好。在 IB 考题中，n 往往直接给出（如“使用 4 个竖条”），或者从等距的数据表中推导出来。在应用复化法则前，务必检查 x 值是否等距。常用符号包括用 x₀, x₁, …, xₙ 表示节点，y₀, y₁, …, yₙ 表示对应的函数值或数据点。

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6. Worked Example with an Integral | 函数积分计算实例

pezoidal rule with n=4 for ∫₁² (1/x) dx, and compare with the exact value ln 2 ≈ 0.693147.

Step 1: Determine h. Here a=1, b=2, so h = (2−1)/4 = 0.25. Nodes: x₀=1, x₁=1.25, x₂=1.5, x₃=1.75, x₄=2.

Step 2: Compute f(x)=1/x at each node.

x	1.0	1.25	1.5	1.75	2.0
f(x)	1.0000	0.8000	0.6667	0.5714	0.5000

Step 3: Apply the composite formula. 2(0.8000+0.6667+0.5714) + 0.5000] = 0.125[1 + 2(2.0381) + 0.5] = 0.125 × 5.5762 = 0.6970.

The trapezoidal approximation is 0.6970, giving an absolute error of about 0.0039, which is less than 0.6%.

用 n=4 的梯形法则近似计算 ∫₁² (1/x) dx，并与精确值 ln 2 ≈ 0.693147 比较。

第 1 步：确定 h。 其中 a=1, b=2，所以 h = (2−1)/4 = 0.25。节点：x₀=1, x₁=1.25, x₂=1.5, x₃=1.75, x₄=2。

第 2 步：计算每个节点处的 f(x)=1/x。

x	1.0	1.25	1.5	1.75	2.0
f(x)	1.0000	0.8000	0.6667	0.5714	0.5000

第 3 步：应用复化公式。 + 2(0.8000+0.6667+0.5714) + 0.5000] = 0.125[1 + 2(2.0381) + 0.5] = 0.125 × 5.5762 = 0.6970。

梯形近似值为 0.6970，绝对误差约为 0.0039，小于 0.6%。

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7. Using a Table of Values | 使用数据表计算

Many IB questions provide a table of measurements rather than an algebraic function. For instance, the velocity v(t) of a car is recorded every 2 seconds:

t (s)	0	2	4	6	8	10
v (m/s)	0	4.2	7.5	9.1	6.3	0

imate the total distance travelled from t=0 to t=10 using the trapezoidal rule. Since h=2,

v₀ + 2(v₁+v₂+v₃+v₄) + v₅] = 2/2 [0 + 2(4.2+7.5+9.1+6.3) + 0] = 1 × [0 + 2(27.1)] = 54.2 m.

The result is 54.2 metres. Always include units and check whether the data points are equally spaced; if not, the composite trapezoidal rule with variable widths must be used.

许多 IB 题目给出的不是代数函数，而是测量数据表。例如，一辆汽车的速度 v(t) 每 2 秒记录一次：

t (s)	0	2	4	6	8	10
v (m/s)	0	4.2	7.5	9.1	6.3	0

用梯形法则估算从 t=0 到 t=10 的总行驶距离。由于 h=2，

距离 [v₀ + 2(v₁+v₂+v₃+v₄) + v₅] = 2/2 [0 + 2(4.2+7.5+9.1+6.3) + 0] = 1 × [0 + 2(27.1)] = 54.2 m。

结果为 54.2 米。始终要带上单位，并检查数据点是否等距；如果不是，则需使用不等距的复合梯形公式。

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8. Error Bounds and Accuracy | 误差界与精度

If f has a continuous second derivative on [a,b], the error E_T in the composite trapezoidal rule satisfies:

|E_T| ≤ (b−a)³/(12 n²) × M, where M = max |f''(x)| on [a,b].

This inequality helps to choose n for a desired accuracy. For the example ∫₁² (1/x) dx, f''(x)=2/x³, whose maximum on [1,2] is 2 at x=1. Thus, with n=4,

|E_T| ≤ 1³/(12×16) × 2 = 2/192 ≈ 0.01042.

hin this bound. Doubling n reduces the error bound by a factor of 4, showing why increasing the number of strips rapidly improves the approximation. The error formula also explains why the trapezoidal rule is exact for linear functions (f'' d to find an upper bound for the error or to determine the smallest n guaranteeing a given tolerance.

如果 f 在 [a,b] 上有连续二阶导数，复化梯形法则的误差 E_T 满足：

|E_T| ≤ (b−a)³/(12 n²) × M，其中 M = max |f''(x)| 在 [a,b] 上。

借助该不等式可根据所需精度选择 n。对于例子 ∫₁² (1/x) dx，f''(x)=2/x³，在 [1,2] 上最大值为 x=1 时的 2。因此，n=4 时，

|E_T| ≤ 1³/(12×16) × 2 = 2/192 ≈ 0.01042。

实际误差为 0.0039，远在界限之内。将 n 加倍可使误差界减至原来的四分之一，这说明增加竖条数目能迅速改善近似。误差公式也解释了为什么梯形法则对线性函数是精确的（f''=0）。考试题目可能会要求计算误差上界，或确定满足给定容差的最小 n 值。

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9. Comparison with Riemann Sums | 与黎曼和的比较

d with L and R, T typically gives a much smaller error, especially for monotonic functions. However, it is usually less accurate than Simpson' e relationships helps in multiple-choice comparisons and when justifying method choices in IB investigations.

梯形法则可视为左黎曼和与右黎曼和的平均值。对于给定的分割，左黎曼和 L = h Σ f(xᵢ₋₁)，右黎曼和 R = h Σ f(xᵢ)，且梯形估计值 T = (L+R)/2。与 L 和 R

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