Definite Integrals: Concepts and Computation Methods | 定积分概念与计算方法

📚 Definite Integrals: Concepts and Computation Methods | 定积分概念与计算方法

The definite integral is one of the central ideas in IB Mathematics. It measures the accumulated change of a function over an interval and has deep connections to area, motion, and probability. This article explains the concept, the Fundamental Theorem of Calculus, and the key techniques you need to compute definite integrals accurately.

定积分是 IB 数学的核心概念之一。它度量函数在一个区间上的累积变化，并与面积、运动和概率紧密相连。这篇文章将解释定积分的概念、微积分基本定理，以及准确计算定积分所需的关键技巧。

1. From Area to Riemann Sums | 从面积到黎曼和

Consider a continuous function f(x) on a closed interval [a, b]. The area under the curve from x = a to x = b can be approximated by dividing the interval into n subintervals of equal width Δx = (b – a)/n, choosing a sample point xᵢ* in each subinterval, and summing f(xᵢ*)Δx. This sum is called a Riemann sum.

考虑一个在闭区间 [a, b] 上的连续函数 f(x)。曲线下从 x = a 到 x = b 的面积可以通过以下方式近似：将区间分成 n 个等宽子区间，宽度 Δx = (b – a)/n，在每个子区间内选取一个采样点 xᵢ*，然后求和 f(xᵢ*)Δx。这个和称为黎曼和。

Sₙ = Σ f(xᵢ*)Δx, Δx = (b – a)/n

If the limit of Sₙ exists as n → ∞ (and Δx → 0), the limit is called the definite integral of f from a to b.

如果当 n → ∞（且 Δx → 0）时 Sₙ 的极限存在，这个极限称为 f 从 a 到 b 的定积分。

The sample point can be the left endpoint, right endpoint, or midpoint of each subinterval; all give the same limit for continuous functions.
采样点可以取每个子区间的左端点、右端点或中点；对于连续函数，所有方式得到相同的极限。
The definite integral is denoted by ∫ₐᵇ f(x) dx.
定积分记作 ∫ₐᵇ f(x) dx。

2. The Definition of the Definite Integral | 定积分的定义

Formally, the definite integral of f from a to b is defined as:

严谨地说，f 从 a 到 b 的定积分定义如下：

∫ₐᵇ f(x) dx = limₙ→∞ Σᵢ₌₁ⁿ f(xᵢ*) Δx

Here a is the lower limit and b is the upper limit. The variable x is a “dummy variable”; we may replace it by any other letter (e.g. t) without changing the value.

这里 a 是积分下限，b 是积分上限。变量 x 是一个”哑变量”；我们可以用任何其他字母（如 t）替换它而不改变值。

Important conventions:

重要约定：

If a = b, then ∫ₐᵃ f(x) dx = 0.
若 a = b，则 ∫ₐᵃ f(x) dx = 0。
If a > b, then ∫ₐᵇ f(x) dx = -∫_bᵃ f(x) dx.
若 a > b，则 ∫ₐᵇ f(x) dx = -∫_bᵃ f(x) dx。

3. Basic Properties of Definite Integrals | 定积分的基本性质

These properties are essential for simplifying integrals and manipulating limits of integration.

这些性质对于简化积分和操作积分限至关重要。

Linearity: ∫ₐᵇ [c₁f(x) + c₂g(x)] dx = c₁∫ₐᵇ f(x) dx + c₂∫ₐᵇ g(x) dx.
线性： ∫ₐᵇ [c₁f(x) + c₂g(x)] dx = c₁∫ₐᵇ f(x) dx + c₂∫ₐᵇ g(x) dx。
Additivity over intervals: If c is in [a,b], then ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫_cᵇ f(x) dx.
区间可加性： 若 c 在 [a,b] 内，则 ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫_cᵇ f(x) dx。
Comparison: If f(x) ≤ g(x) on [a,b], then ∫ₐᵇ f(x) dx ≤ ∫ₐᵇ g(x) dx.
比较性质： 若在 [a,b] 上有 f(x) ≤ g(x)，则 ∫ₐᵇ f(x) dx ≤ ∫ₐᵇ g(x) dx。

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

The Fundamental Theorem of Calculus (FTC) connects differentiation and integration. It has two parts:

微积分基本定理（FTC）将微分和积分联系起来。它包含两个部分：

Part 1: If F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) – F(a).

第一部分： 若 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) – F(a)。

Part 2: If g(x) = ∫ₐˣ f(t) dt, then g'(x) = f(x).

第二部分： 若 g(x) = ∫ₐˣ f(t) dt，则 g'(x) = f(x)。

The notation F(b) – F(a) is often abbreviated as [F(x)]ₐᵇ or F(x)|ₐᵇ.

记号 F(b) – F(a) 通常缩写为 [F(x)]ₐᵇ 或 F(x)|ₐᵇ。

5. Computing Directly by Finding Antiderivatives | 通过求原函数直接计算

The simplest way to evaluate a definite integral is to find an antiderivative and then substitute the limits.

计算定积分最简单的方法是求出原函数，然后代入上下限。

Example: Evaluate ∫₁³ (2x + 1) dx.

示例：计算 ∫₁³ (2x + 1) dx。

∫₁³ (2x + 1) dx = [x² + x]₁³ = (9 + 3) – (1 + 1) = 10

Common antiderivatives you must know:

你必须掌握的常见原函数：

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C (for n ≠ -1)
∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C
∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C
∫ sec²x dx = tan x + C

6. Substitution (u-substitution) | 换元法（u 代换）

When the integrand contains a composite function, substitute a new variable to simplify the integral. Remember to change the limits of integration when using definite integrals.

当被积函数包含复合函数时，可以引入新变量来简化积分。使用定积分时，记得要改变积分上下限。

Procedure:

步骤：

Let u = g(x), then du = g'(x) dx.
令 u = g(x)，则 du = g'(x) dx。
Transform the limits: when x = a, u = g(a); when x = b, u = g(b).
转换上下限：当 x = a 时，u = g(a)；当 x = b 时，u = g(b)。
Evaluate ∫_{g(a)}^{g(b)} f(u) du.
计算 ∫_{g(a)}^{g(b)} f(u) du。

Example: ∫₀¹ 2x e^{x²} dx. Let u = x², du = 2x dx. When x = 0, u = 0; x = 1, u = 1. So the integral becomes ∫₀¹ eᵘ du = e – 1.

示例：∫₀¹ 2x e^{x²} dx。令 u = x²，du = 2x dx。当 x = 0 时，u = 0；x = 1 时，u = 1。于是积分变为 ∫₀¹ eᵘ du = e – 1。

7. Integration by Parts | 分部积分法

Integration by parts is used for products of functions. The formula comes from the product rule for derivatives.

分部积分法用于函数乘积的积分。该公式来源于乘积法则。

∫ₐᵇ u dv = [uv]ₐᵇ – ∫ₐᵇ v du

Choose u that simplifies when differentiated (e.g. polynomials), and dv that is easy to integrate (e.g. eˣ dx, sin x dx).

选择微分后会简化的 u（例如多项式），以及容易积分的 dv（例如 eˣ dx、sin x dx）。

Example: ∫₀¹ x eˣ dx. Let u = x, dv = eˣ dx. Then du = dx, v = eˣ. So the integral equals [x eˣ]₀¹ – ∫₀¹ eˣ dx = e – (e – 1) = 1.

示例：∫₀¹ x eˣ dx。令 u = x，dv = eˣ dx。则 du = dx，v = eˣ。因此积分等于 [x eˣ]₀¹ – ∫₀¹ eˣ dx = e – (e – 1) = 1。

8. Using Symmetry: Odd and Even Functions | 利用对称性：奇函数与偶函数

Symmetry can dramatically simplify definite integrals over symmetric intervals [-a, a].

对称性可以大大简化对称区间 [-a, a] 上的定积分。

If f is even: f(-x) = f(x), then ∫_{-a}^{a} f(x) dx = 2∫₀ᵃ f(x) dx.
若 f 是偶函数：f(-x) = f(x)，则 ∫_{-a}^{a} f(x) dx = 2∫₀ᵃ f(x) dx。
If f is odd: f(-x) = -f(x), then ∫_{-a}^{a} f(x) dx = 0.
若 f 是奇函数：f(-x) = -f(x)，则 ∫_{-a}^{a} f(x) dx = 0。

Example: ∫_{-π}^{π} x³ cos x dx = 0 because x³ cos x is odd (odd times even is odd).

示例：∫_{-π}^{π} x³ cos x dx = 0，因为 x³ cos x 是奇函数（奇函数乘偶函数仍是奇函数）。

9. Definite Integrals and Area | 定积分与面积

The geometric interpretation of ∫ₐᵇ f(x) dx is the signed area between the curve and the x-axis. Areas above the axis count as positive, and areas below the axis count as negative.

∫ₐᵇ f(x) dx 的几何意义是曲线与 x 轴之间的有向面积。轴上方的面积记为正，轴下方的面积记为负。

To find the total (unsigned) area between a curve and the x-axis, you must split the interval at zeros of f and take absolute values.

要求曲线与 x 轴之间的总（无符号）面积，你需要在 f 的零点处拆分区间，并取绝对值。

Example: f(x) = x² – 1 on [-2, 2]. The integral is zero by symmetry, but the total area is 2∫₁² (x² – 1) dx + 2∫₀¹ (1 – x²) dx = 4/3 + 4/3 = 8/3.

示例：f(x) = x² – 1 在 [-2, 2] 上。由对称性知定积分为 0，但总面积为 2∫₁² (x² – 1) dx + 2∫₀¹ (1 – x²) dx = 4/3 + 4/3 = 8/3。

10. Improper Integrals | 反常积分

An integral is improper if one or both limits are infinite, or if the integrand has an infinite discontinuity within the interval.

如果积分限有一个或两个为无穷大，或者被积函数在区间内有无穷间断点，那么这个积分就是反常积分。

For infinite limits:

对于无穷限：

∫ₐ^∞ f(x) dx = lim_{b→∞} ∫ₐᵇ f(x) dx

The integral converges if the limit exists; otherwise it diverges.

如果极限存在，则积分收敛；否则发散。

Example: ∫₁^∞ 1/x² dx = lim_{b→∞} [ -1/x ]₁ᵇ = 1. It converges.

示例：∫₁^∞ 1/x² dx = lim_{b→∞} [ -1/x ]₁ᵇ = 1。它收敛。

11. Common Pitfalls and Tips | 常见陷阱与技巧

Many marks are lost in IB exams due to small errors. Keep these in mind:

在 IB 考试中，很多分数因为小错误而丢失。请记住以下几点：

When using substitution in a definite integral, always update the limits; do not substitute back after integrating.
在定积分中使用换元法时，务必更新上下限；积分后不必代回原变量。
Check whether the function is odd or even before integrating on a symmetric interval.
在对称区间上积分之前，先检查函数是奇函数还是偶函数。
For rational functions, consider partial fractions before integrating.
对有理函数，先考虑部分分式再积分。
Be careful with absolute values when integrating 1/x over negative intervals.
在负区间上积分 1/x 时，注意绝对值。

12. Summary | 总结

You should now understand the definition of the definite integral as a limit of Riemann sums, the role of the Fundamental Theorem of Calculus, and the main computation methods: direct antiderivatives, substitution, integration by parts, and symmetry arguments. Practice these techniques with past paper questions to build confidence.

你现在应该理解了定积分作为黎曼和极限的定义、微积分基本定理的作用，以及主要的计算方法：直接求原函数、换元法、分部积分法和对称性讨论。通过练习历年真题来掌握这些技巧，并建立信心。

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Definite Integrals: Concepts and Computation Methods | 定积分概念与计算方法