A-Level Mathematics: Definite Integrals – Concepts and Computation | A-Level 数学：定积分的概念与计算

📚 A-Level Mathematics: Definite Integrals – Concepts and Computation | A-Level 数学：定积分的概念与计算

In A-Level mathematics, the definite integral is one of the most important tools in calculus. It allows you to calculate the exact accumulation of a quantity, such as area, displacement, or total change, over a finite interval. The definite integral of a function f(x) from x = a to x = b is written as ∫_a^b f(x) dx.

在 A-Level 数学中，定积分是微积分最重要的工具之一。它能让你精确计算某个量在一个有限区间内的累积量，比如面积、位移或总变化量。函数 f(x) 从 x = a 到 x = b 的定积分写作 ∫_a^b f(x) dx。

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1. What Does a Definite Integral Represent? | 定积分表示什么？

Geometrically, the definite integral measures the signed area between the curve y = f(x) and the x-axis over the interval [a, b]. The word “signed” means that areas above the x-axis contribute positive values, while areas below the x-axis contribute negative values.

从几何上看，定积分衡量的是曲线 y = f(x) 与 x 轴在区间 [a, b] 之间的有向面积。“有向”意味着 x 轴上方的面积贡献为正值，而下方的面积贡献为负值。

If f(x) = 0 everywhere, the integral is zero. If f(x) is positive on the whole interval, the definite integral equals the ordinary geometric area. If f(x) is negative on the whole interval, the definite integral equals the negative of that area.

若 f(x) 处处等于 0，则定积分为 0。若 f(x) 在整个区间上恒为正，则定积分等于普通几何面积；若 f(x) 在整个区间上恒为负，则定积分等于该面积的相反数。

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2. Riemann Sums: The Limit Definition | 黎曼和：极限定义

To define ∫_a^b f(x) dx rigorously, we divide the interval [a, b] into n subintervals of equal width Δx = (b – a)/n. On each subinterval we choose a sample point xᵢ*, form the product f(xᵢ*) Δx, and sum these products from i = 1 to n.

为了严格定义 ∫_a^b f(x) dx，我们将区间 [a, b] 分成 n 个等宽子区间，宽度为 Δx = (b – a)/n。在每个子区间内取一个样本点 xᵢ*，构成乘积 f(xᵢ*) Δx，并将这些乘积从 i = 1 到 n 相加。

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

When n tends to infinity and this limit exists, it defines the definite integral of f over [a, b]. In A-Level problems you rarely evaluate this limit algebraically, but understanding the definition helps you interpret the meaning of every integral.

当 n 无限增大且该极限存在时，这个极限就定义为 f 在 [a, b] 上的定积分。在 A-Level 题目中，你很少需要从代数上直接求这个极限，但理解这一定义能帮助你搞清楚每个积分的含义。

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

The fundamental theorem of calculus is the bridge between differentiation and integration. If f is continuous on [a, b] and F is any antiderivative of f, then the definite integral can be evaluated simply by finding F(b) – F(a).

微积分基本定理是微分与积分之间的桥梁。若 f 在 [a, b] 上连续，F 是 f 的任意一个原函数，那么定积分可以通过计算 F(b) – F(a) 来求得。

∫_a^b f(x) dx = F(b) – F(a)

Another version of the theorem states that the derivative of an integral with a variable upper limit recovers the original integrand:

该定理的另一个版本指出：变上限积分的导数会恢复原来的被积函数：

d/dx ∫_a^x f(t) dt = f(x)

This result is frequently tested in A-Level examination questions, especially when combined with the chain rule.

这一结论在 A-Level 考试中经常出现，尤其是与链式法则结合的时候。

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

Several properties make definite integrals easier to manipulate. You should be able to apply all of them quickly and confidently.

以下性质能让定积分的运算更加方便。你应该能够快速且自信地运用它们。

• Linearity / 线性性： ∫_a^b [c f(x) + d g(x)] dx = c ∫_a^b f(x) dx + d ∫_a^b g(x) dx, where c and d are constants.
• Reversing limits / 反转上下限： ∫_a^b f(x) dx = -∫_b^a f(x) dx.
• Additivity of intervals / 区间可加性： ∫_a^c f(x) dx + ∫_c^b f(x) dx = ∫_a^b f(x) dx.
• Zero width / 零宽度区间： ∫_a^a f(x) dx = 0.
• Comparison property / 比较性质： If f(x) ≤ g(x) on [a, b], then ∫_a^b f(x) dx ≤ ∫_a^b g(x) dx.

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5. Evaluating Definite Integrals Directly | 直接计算定积分

The quickest way to evaluate a definite integral is to find an antiderivative, write it inside square brackets, then substitute the upper limit and the lower limit, and subtract the lower value from the upper value.

计算定积分最快的方法是先求一个原函数，把它写在方括号中，然后分别代入上限和下限，并用上限处的值减去下限处的值。

∫₁² x² dx = [x³/3]₁² = (8/3) – (1/3) = 7/3

Notice that the constant of integration is not needed for a definite integral. When you evaluate F(b) – F(a), any arbitrary constant cancels out automatically.

注意，定积分不需要加积分常数 C。当你计算 F(b) – F(a) 时，任意常数会自动抵消。

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6. Integration by Substitution | 换元积分法

Substitution simplifies an integral by introducing a new variable. For definite integrals, you can either change the limits to the new variable and never convert back, or integrate in the new variable and then convert back before substituting the original limits. The first method is usually cleaner.

换元积分法通过引入新变量来化简积分。对于定积分，你可以选择将上限和下限改为新变量对应的值，并且不再换回原变量；也可以先用新变量积分，再换回原变量，最后代入原来的上下限。通常第一种方法更简洁。

For example, evaluate ∫₀¹ 2x e^x² dx. Let u = x², so du = 2x dx. As x increases from 0 to 1, u also increases from 0 to 1. Therefore:

例如，计算 ∫₀¹ 2x e^x² dx。令 u = x²，则 du = 2x dx。当 x 从 0 增大到 1 时，u 也从 0 增大到 1。因此：

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