Definite Integrals | 定积分

📚 Definite Integrals | 定积分

In the IB Mathematics curriculum, definite integrals are a fundamental component of calculus, allowing us to calculate exact areas, volumes, and total change. Mastery of the definition, properties, and evaluation techniques—along with the connection to the Fundamental Theorem of Calculus—is essential for both Analysis & Approaches and Applications & Interpretation courses. This article provides a thorough revision of definite integrals, covering core concepts and common applications.

在 IB 数学课程中，定积分是微积分的核心组成部分，使我们能够计算精确的面积、体积和总变化量。掌握其定义、性质、计算技巧以及与微积分基本定理的联系，对于分析与方法和应用与解释两门课程都至关重要。本文将对定积分进行全面复习，涵盖核心概念和常见应用。

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1. Definition and Notation | 定义与符号

The definite integral of a continuous function f(x) from a to b is written as ∫_a^b f(x) dx. It represents the net signed area between the curve y = f(x) and the x-axis from x = a to x = b. Regions above the x-axis contribute positive area, while regions below contribute negative area. The numbers a and b are called the lower and upper limits of integration, and dx indicates that the integration is performed with respect to the variable x.

连续函数 f(x) 从 a 到 b 的定积分记作 ∫_a^b f(x) dx。它表示曲线 y = f(x) 与 x 轴在 x = a 和 x = b 之间的净有号面积。位于 x 轴上方的区域贡献正面积，位于下方的区域贡献负面积。a 和 b 分别称为积分下限和上限，dx 表示积分变量为 x。

A crucial difference from an indefinite integral is that a definite integral yields a number rather than a family of functions. The integral sign ∫ evolved from the long letter ‘S’, standing for ‘summation’, which reflects its origin in Riemann sums.

与不定积分的一个重要区别是，定积分的结果是一个数值而不是函数族。积分号 ∫ 由长字母 ‘S’ 演变而来，代表“求和”，这反映了它源自黎曼和的背景。

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2. Riemann Sums and the Limit | 黎曼和与极限

The formal definition of a definite integral is based on Riemann sums. We start by partitioning [a, b] into n subintervals of equal width Δx = (b − a)/n. In each subinterval we choose a sample point x_i^*, then form the sum Σ_i=1ⁿ f(x_i^*) Δx. The definite integral is the limit of these sums as n approaches infinity, provided the limit exists and is the same for any choice of sample points.

定积分的正式定义建立在黎曼和的基础上。我们将 [a, b] 等分为 n 个宽度为 Δx = (b − a)/n 的子区间，在每个子区间内选取一个样本点 x_i^*，然后构造和式 Σ_i=1ⁿ f(x_i^*) Δx。当 n 趋于无穷大时，这些和的极限就是定积分，前提是该极限存在且与样本点的选取无关。

∫_a^b f(x) dx = lim_n→∞ Σ_i=1ⁿ f(x_i^*) Δx

In the IB context, you are not required to evaluate limits of Riemann sums directly; instead, you use the Fundamental Theorem of Calculus. However, understanding the Riemann sum definition helps to interpret the definite integral as an accumulator of infinitesimal products.

在 IB 课程中，不要求大家直接计算黎曼和的极限，而是借助微积分基本定理。然而，理解黎曼和定义有助于将定积分理解为无穷小乘积的累加器。

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3. Fundamental Theorem of Calculus | 微积分基本定理

The Fundamental Theorem of Calculus (FTC) bridges differentiation and integration. It consists of two parts. Part 1 states: if F is any antiderivative of f on [a, b], then ∫_a^b f(x) dx = F(b) − F(a). This simple formula allows us to evaluate definite integrals by finding an antiderivative and substituting the limits. Part 2 deals with the accumulation function A(x) = ∫_a^x f(t) dt and shows that A′(x) = f(x).

微积分基本定理（FTC）将微分与积分联系起来。它包含两个部分。第一部分：若 F 是 f 在 [a, b] 上的任一原函数，则 ∫_a^b f(x) dx = F(b) − F(a)。这个简洁的公式使我们能够通过寻找原函数并代入上下限来计算定积分。第二部分涉及累积函数 A(x) = ∫_a^x f(t) dt，并表明 A′(x) = f(x)。

∫_a^b f(x) dx = F(b) − F(a),   where F′(x) = f(x)

When writing your solution, it is common to show the antiderivative in square brackets with the limits: [F(x)]_a^b = F(b) − F(a). This notation helps avoid algebraic mistakes when evaluating trigonometric or logarithmic functions.

解题时，通常会将原函数写在带上下限的方括号内：[F(x)]_a^b = F(b) − F(a)。这种记法有助于在计算三角函数或对数函数时避免代数错误。

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4. Properties of Definite Integrals | 定积分的性质

Definite integrals obey several algebraic rules that simplify calculations and help in splitting the area over intervals. The following properties are fundamental and appear frequently in IB problems:

定积分满足若干代数运算法则，这些法则可以简化计算并帮助在不同区间上分割面积。以下基本性质在 IB 问题中经常出现：

• Constant multiple: ∫_a^b k f(x) dx = k ∫_a^b f(x) dx

  常数倍：∫_a^b k f(x) dx = k ∫_a^b f(x) dx

• Sum/difference: ∫_a^b [f(x) ± g(x)] dx = ∫_a^b f(x) dx ± ∫_a^b g(x) dx

  和/差：∫_a^b [f(x) ± g(x)] dx = ∫_a^b f(x) dx ± ∫_a^b g(x) dx

• Additivity over intervals: ∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx for any c between a and b

  区间可加性：∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx，其中 c 介于 a 与 b 之间

• Reversal of limits: ∫_a^b f(x) dx = −∫_b^a f(x) dx

  上下限对换：∫_a^b f(x) dx = −∫_b^a f(x) dx

• Zero width: ∫_a^a f(x) dx = 0

  零宽度：∫_a^a f(x) dx = 0

These properties are particularly useful when a graph consists of different pieces or when symmetry allows simplification, such as integrating an odd function over a symmetric interval.

当图形由不同部分组成，或者利用对称性（例如奇函数在对称区间上积分）可以化简时，这些性质尤其有用。

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5. Evaluating Definite Integrals | 计算定积分

To evaluate a definite integral analytically, first find an antiderivative F(x) of the integrand f(x). Then substitute the upper and lower limits and subtract: F(b) − F(a). You must be comfortable with antiderivatives of polynomials, exponentials, logarithms, and trigonometric functions, as well as standard forms like ∫ 1/x dx = ln|x| + C.

要解析地计算定积分，首先求出被积函数 f(x) 的一个原函数 F(x)，然后代入上下限并相减：F(b) − F(a)。你必须熟练掌握多项式、指数函数、对数函数和三角函数的原函数，以及如 ∫ 1/x dx = ln|x| + C 的标准形式。

Always remember to use radians for trigonometric functions when performing calculus in IB. When evaluating, it is good practice to simplify the antiderivative as much as possible before plugging in the limits to reduce arithmetic errors. For example, ∫₀¹ (3x² + 2) dx = [x³ + 2x]₀¹ = (1+2) − (0) = 3.

请务必记住，在 IB 微积分中，三角函数的变量均采用弧度制。计算时，最好在代入上下限之前尽量化简原函数，以减少算术错误。例如，∫₀¹ (3x² + 2) dx = [x³ + 2x]₀¹ = (1+2) − (0) = 3。

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6. Integration by Substitution for Definite Integrals | 定积分的换元积分法

When evaluating ∫_a^b f(g(x)) g′(x) dx, substitution simplifies the integrand. Let u = g(x), then du = g′(x) dx. Crucially, you must change the limits of integration: when x = a, u = g(a); when x = b, u = g(b). The integral becomes ∫_g(a)^g(b) f(u) du, which can be evaluated using the FTC.

计算 ∫_a^b f(g(x)) g′(x) dx 时，换元法可以化简被积函数。令 u =

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