Definite Integrals | 定积分

📚 Definite Integrals | 定积分

In A-Level mathematics, a definite integral is the signed area between a curve and the x-axis over a closed interval [a, b]. The Fundamental Theorem of Calculus links the definite integral to an antiderivative, making it possible to evaluate integrals exactly without summing infinitely many rectangles.

在 A-Level 数学中，定积分表示曲线与 x 轴在闭区间 [a, b] 上的有向面积。微积分基本定理将定积分与原函数联系起来，使我们无需无限求和便能精确计算积分。

1. From Indefinite to Definite: The Fundamental Theorem | 从不定积分到定积分：微积分基本定理

If F(x) is an antiderivative of f(x) on [a, b], then the definite integral of f from a to b is given by the formula ∫ₐᵇ f(x) dx = F(b) − F(a).

如果 F(x) 是 f(x) 在 [a, b] 上的一个原函数，那么 f 从 a 到 b 的定积分由公式 ∫ₐᵇ f(x) dx = F(b) − F(a) 给出。

This result is central to Edexcel Pure Mathematics: once you find an indefinite integral, evaluation becomes a simple substitution of the upper and lower limits followed by subtraction.

这一结果是 Edexcel 纯数学的核心：一旦找到不定积分，计算就变成代入上限和下限并求差。

For example, ∫₁² 3x² dx = [x³]₁² = 2³ − 1³ = 7.

例如，∫₁² 3x² dx = [x³]₁² = 2³ − 1³ = 7。

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2. Notation and Limits of Integration | 定积分符号与积分限

In the symbol ∫ₐᵇ f(x) dx, the lower limit is a and the upper limit is b. The variable x is a dummy variable, so ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt.

在符号 ∫ₐᵇ f(x) dx 中，下限为 a，上限为 b。变量 x 是哑变量，因此 ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt。

The value of a definite integral is a number, not a family of functions. It does not include the constant of integration because that constant cancels out when subtracting F(a) from F(b).

定积分的值是一个数，而不是一族函数。它不包含积分常数，因为在 F(b) − F(a) 的相减中常数会抵消。

You should always write the limits carefully alongside the integral sign, especially when the integrand contains more than one variable or parameter.

在积分号旁一定要仔细写出积分限，尤其当被积函数含有多个变量或参数时更应如此。

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3. Key Properties of Definite Integrals | 定积分的重要性质

Definite integrals obey linearity: for constants c and d, ∫ₐᵇ [c f(x) + d g(x)] dx = c∫ₐᵇ f(x) dx + d∫ₐᵇ g(x) dx.

定积分满足线性性质：对任意常数 c 和 d，有 ∫ₐᵇ [c f(x) + d g(x)] dx = c∫ₐᵇ f(x) dx + d∫ₐᵇ g(x) dx。

They also obey interval additivity: if c lies between a and b, then ∫ₐᶜ f(x) dx + ∫꜀ᵇ f(x) dx = ∫ₐᵇ f(x) dx.

它们还满足区间可加性：若 c 在 a 与 b 之间，则 ∫ₐᶜ f(x) dx + ∫꜀ᵇ f(x) dx = ∫ₐᵇ f(x) dx。

These properties allow you to split complicated integrals into smaller pieces and to handle absolute value or piecewise functions in a systematic way.

这些性质使你可以将复杂积分拆分成较小的部分，并系统地处理绝对值函数或分段函数。

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4. Reversing Limits and Zero Width | 交换积分限与零宽度

Reversing the limits changes the sign of a definite integral: the integral from b to a is −∫ₐᵇ f(x) dx. This follows directly from F(a) − F(b) = −[F(b) − F(a)].

交换积分限会改变定积分的符号：从 b 到 a 的积分等于 −∫ₐᵇ f(x) dx。这直接源于 F(a) − F(b) = −[F(b) − F(a)]。

If the upper and lower limits are equal, the integral is zero: ∫ₐᵃ f(x) dx = 0, since F(a) − F(a) = 0.

若上下限相等，则定积分为零：∫ₐᵃ f(x) dx = 0，因为 F(a) − F(a) = 0。

In exam questions, be careful not to assume ∫ₐᵇ f(x) dx = ∫ᵇ? f(x) dx without changing sign; reversing limits is one of the most common error points.

在考试题中，注意不要把从 a 到 b 的积分与交换限后的积分直接相等；交换积分限会改变符号，这是最常见的易错点之一。

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5. Evaluating by Substitution | 换元法计算定积分

When using substitution u = g(x) in a definite integral, you can either change the limits as you substitute or evaluate the indefinite integral first and then substitute back. For Edexcel, changing limits is usually quicker and avoids reintroducing x.

在定积分中使用换元 u = g(x) 时，可以随换元改变积分限，也可以先求不定积分再代回原变量。就 Edexcel 而言，改变积分限通常更快捷，也避免重新引入 x。

For example, consider ∫₀¹ 2x(x² + 1)³ dx. Let u = x² + 1, so du = 2x dx. When x = 0, u = 1; when x = 1, u = 2. The integral becomes ∫₁² u³ du = [u⁴/4]₁² = 4 − 1/4 = 15/4.

例如，考虑 ∫₀¹ 2x(x² + 1)³ dx。设 u = x² + 1，则 du = 2x dx。当 x = 0 时 u = 1；当 x = 1 时 u = 2。积分变为 ∫₁² u³ du = [u⁴/4]₁² = 4 − 1/4 = 15/4。

When changing limits, always state the new limits clearly; many students forget to convert the limits and leave the original x-values in the u-integral.

改变积分限时，一定要清楚写出新的上下限；许多学生忘记转换积分限，而在 u 的积分中仍保留原来的 x 值。

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6. Integration by Parts with Limits | 带限的分部积分

For a definite integral, integration by parts uses the same formula with limits applied to the uv term: ∫ₐᵇ u dv = [uv]ₐᵇ − ∫ₐᵇ v du.

对定积分应用分部积分公式时，只需将上下限代入 uv 项：∫ₐᵇ u dv = [uv]ₐᵇ − ∫ₐᵇ v du。

Example: ∫₀¹ x e

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