Definite Integral | 定积分

📚 Definite Integral | 定积分

The definite integral is a central concept in calculus, representing the accumulation of quantities and the area under a curve over a specific interval. In IB Mathematics, you learn to interpret it as the limit of Riemann sums and apply the Fundamental Theorem of Calculus to evaluate it efficiently. This article revises key definitions, properties, evaluation techniques, and applications such as area calculations, numerical integration, and improper integrals.

定积分是微积分的核心概念，表示在一个特定区间上量的累积和曲线下方的面积。在IB数学中，你要学会将定积分理解为黎曼和的极限，并运用微积分基本定理高效地求值。本文复习关键定义、性质、计算技巧以及面积计算、数值积分和反常积分等应用。

1. Introduction to Definite Integrals | 定积分简介

The definite integral, denoted ∫_a^b f(x) dx, computes the net area between the graph of a function and the x‑axis from x = a to x = b. It arises naturally in problems of distance, volume, and total growth. Unlike an indefinite integral (antiderivative), a definite integral yields a numerical value, which can be positive, negative, or zero depending on the position of the curve relative to the x‑axis.

定积分记作 ∫_a^b f(x) dx，计算函数图像与 x 轴之间从 x = a 到 x = b 的净面积。它自然地出现在距离、体积和总量增长问题中。与不定积分（原函数）不同，定积分给出一个数值，该数值根据曲线相对于 x 轴的位置可正、可负或为零。

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

The definite integral is defined as the limit of Riemann sums. Partition the interval [a, b] into n subintervals of equal width Δx = (b − a)/n. Choose a sample point x_i* in each subinterval (e.g. left endpoint, right endpoint, or midpoint). The Riemann sum is Σ_i=1ⁿ f(x_i*) Δx. As n → ∞, this sum approaches the exact integral, provided the limit exists.

定积分被定义为黎曼和的极限。将区间 [a, b] 分成 n 个等宽子区间，宽度为 Δx = (b − a)/n。在每个子区间中选取样本点 x_i*（例如左端点、右端点或中点）。黎曼和为 Σ_i=1ⁿ f(x_i*) Δx。当 n → ∞ 时，若极限存在，该和式趋向精确的积分值。

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

Continuous functions on a closed interval are always integrable, but piecewise functions with finitely many jumps are also integrable. Riemann sums form the theoretical basis and are used in numerical methods.

闭区间上的连续函数总是可积的，而只有有限个跳跃间断点的分段函数也可积。黎曼和构成了理论基础，并在数值方法中使用。

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

The Fundamental Theorem of Calculus (FTC) bridges differentiation and integration. The part most used for evaluation states: if F is any antiderivative of f on [a, b] (i.e., F ‘(x) = f(x)), then ∫_a^b f(x) dx = F(b) − F(a). This transforms the limit definition into a straightforward calculation. The second part of FTC gives the derivative of an integral with a variable upper limit: d/dx ∫_a^x f(t) dt = f(x).

微积分基本定理（FTC）连接了微分与积分。最常用于求值的部分指出：若 F 是 f 在 [a, b] 上的任意一个原函数（即 F ‘(x) = f(x)），则 ∫_a^b f(x) dx = F(b) − F(a)。这将极限定义转化为直接的计算。FTC 的第二部分给出了变上限积分的导数：d/dx ∫_a^x f(t) dt = f(x)。

For example, to find ∫₀² 3x² dx, note that an antiderivative of 3x² is x³. Then the integral equals [x³]₀² = 8 − 0 = 8. No constant of integration is needed because it cancels out.

例如，求 ∫₀² 3x² dx，注意到 3x² 的一个原函数是 x³。于是积分等于 [x³]₀² = 8 − 0 = 8。不需要加积分常数，因为它会抵消。

4. Evaluating Definite Integrals – Step by Step | 逐步计算定积分

To evaluate a definite integral analytically:

• Find an antiderivative F(x) of f(x) (indefinite integration).
• Substitute the upper limit: F(b).
• Substitute the lower limit: F(a).
• Subtract: F(b) − F(a).

For integrals requiring substitution, remember to change the limits or substitute back before applying the limits. For example, ∫₀¹ 2x√(x²+1) dx. Let u = x²+1, then du = 2x dx. When x = 0, u = 1; when x = 1, u = 2. The integral becomes ∫₁² √u du = [2/3 u^3/2]₁².

要解析地计算定积分：先求 f(x) 的原函数 F(x)（不定积分），代入上限得 F(b)，代入下限得 F(a)，然后相减：F(b) − F(a)。对于需要换元的积分，记得改变积分限或者在代回原变量后再代入上下限。例如，∫₀¹ 2x√(x²+1) dx。令 u = x²+1，则 du = 2x dx。当 x = 0 时 u = 1；当 x = 1 时 u = 2。积分变为 ∫₁² √u du = [2/3 u^3/2]₁²。

A common example using trigonometric antiderivatives: ∫₀^π sin x dx = [−cos x]₀^π = (−cos π) − (−cos 0) = (1) − (−1) = 2.

一个使用三角函数原函数的常见例子：∫₀^π sin x dx = [−cos x]₀^π = (−cos π) − (−cos 0) = 1 − (−1) = 2。

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

Definite integrals have a set of algebraic properties that simplify calculations and proofs. The table below summarises the most important ones.

定积分有一组代数性质，可以简化计算与证明。下表总结了最重要的性质。

These properties are especially useful when splitting integrals across singularities, handling piecewise functions, or estimating integrals without computing them exactly.

这些性质在跨越奇点分割积分、处理分段函数，或者不精确计算而估计积分时尤其有用。

6. Area Between Curves | 曲线之间的面积

The definite integral ∫_a^b f(x) dx gives signed area. For the actual geometric area between two curves y = f(x) and y = g(x) from a to b, we integrate the absolute difference: ∫_a^b |f(x) − g(x)| dx. If one function is always above the other, say f(x) ≥ g(x) on [a,b], the area simplifies to ∫_a^b [f(x) − g(x)] dx. When the curves intersect, split the integral at the intersection points, using the top-minus-bottom rule in each subinterval.

定积分 ∫_a^b f(x) dx 给出的是带符号的面积。对于两条曲线 y = f(x) 与 y = g(x) 之间从 a 到 b 的实际几何面积，我们对绝对差进行积分：∫_a^b |f(x) − g(x)| dx。若一个函数始终在另一个之上，比如在 [a,b] 上 f(x) ≥ g(x)，面积简化为 ∫_a^b [f(x) − g(x)] dx。当曲线相交时，以交点为界分割积分，在每个子区间用“上减下”的规则。

Example: Find the area enclosed by y = x² and y = x. Solve x² = x ⇒ x = 0, 1. On [0,1], x ≥ x². The area = ∫₀¹ (x − x²) dx = [½ x² − ⅓ x³]_{Published by TutorHao | IB Mathematics Revision Series | aleveler.com}

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