Definite Integration: Methods and Applications | 定积分的计算方法与应用

📚 Definite Integration: Methods and Applications | 定积分的计算方法与应用

The definite integral is a cornerstone of A-Level Mathematics. It allows us to calculate exact areas, volumes, average values, and many real-world quantities that vary continuously. This article reviews the key techniques for evaluating definite integrals and explores their most important applications in the Edexcel syllabus.

定积分是A-Level数学的基石。它可以精确计算面积、体积、平均值以及许多随连续变化而变化的实际量。本文将回顾求解定积分的关键技巧，并探讨其在Edexcel考纲中最重要的应用。

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

If f(x) is continuous on [a, b] and F(x) is an antiderivative of f(x), then the definite integral of f from a to b is given by F(b) − F(a). This theorem links differentiation and integration, turning the problem of finding an area into evaluating a function difference.

若 f(x) 在 [a, b] 上连续，且 F(x) 是 f(x) 的一个原函数，则 f 从 a 到 b 的定积分等于 F(b) − F(a)。这一基本定理将微分与积分联系起来，把求面积问题转化为计算函数值之差。

∫ₐᵇ f(x) dx = F(b) − F(a)

Always remember that the result of a definite integral is a number, not a function.

切记：定积分的结果是一个数值，而不是一个函数。

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

Several properties simplify evaluation and manipulation. These appear throughout exam questions and are essential for restructuring integrals.

定积分有几条常用性质，能简化计算与变形，也是考试中重组积分表达式的关键。

Linearity: ∫ₐᵇ [f(x) ± g(x)] dx = ∫ₐᵇ f(x) dx ± ∫ₐᵇ g(x) dx; and ∫ₐᵇ k·f(x) dx = k·∫ₐᵇ f(x) dx.

线性性质：∫ₐᵇ [f(x) ± g(x)] dx = ∫ₐᵇ f(x) dx ± ∫ₐᵇ g(x) dx；并且 ∫ₐᵇ k·f(x) dx = k·∫ₐᵇ f(x) dx。
Interval additivity: ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫꜀ᵇ f(x) dx for any c between a and b.

区间可加性：若 c 介于 a 和 b 之间，则 ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫꜀ᵇ f(x) dx。
Swapping limits changes the sign: ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx.

交换上下限改变符号：∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx。
Zero-width integral: ∫ₐᵃ f(x) dx = 0.

宽度为零的积分：∫ₐᵃ f(x) dx = 0。

3. Substitution Method | 换元积分法

Many integrals become simpler after a change of variable. For a definite integral, the limits must also be changed when the variable is changed.

许多积分通过变量替换会变得简单。对于定积分，换元时必须同步更改积分上下限。

∫ₐᵇ f(g(x))·g'(x) dx = ∫₍g(a)₎⁽ᵍ⁽ᵇ⁾⁾ f(u) du

For example, to evaluate ∫₀¹ 2x·e^(x²) dx, set u = x², so du = 2x dx. The new limits are u(0)=0 and u(1)=1. The integral becomes ∫₀¹ eᵘ du = e − 1.

例如，求 ∫₀¹ 2x·e^(x²) dx，令 u = x²，则 du = 2x dx。新限为 u(0)=0 和 u(1)=1，积分变为 ∫₀¹ eᵘ du = e − 1。

In Edexcel exams, always show the substituted limits explicitly—this avoids errors and earns method marks.

在 Edexcel 考试中，务必明确写出替换后的上下限，这有助于避免错误并获得方法分。

4. Integration by Parts | 分部积分法

When the integrand is a product of two functions, integration by parts is often effective. For definite integrals, apply the formula and then evaluate each term at the limits.

当被积函数是两个函数之积时，分部积分法常常奏效。对定积分，先套用公式，再在各限处取值。

∫ₐᵇ u dv = [u·v]ₐᵇ − ∫ₐᵇ v du

Choose u to be the function that simplifies when differentiated (e.g. x, ln x) and dv to be the part that is easy to integrate (e.g. eˣ, sin x).

选取 u 为求导后会简化的函数（如 x、ln x），选取 dv 为易于积分的部分（如 eˣ、sin x）。

Example: ∫₀¹ x·eˣ dx. Take u = x, dv = eˣ dx, so du = dx, v = eˣ. Then [x·eˣ]₀¹ − ∫₀¹ eˣ dx = e − (e − 1) = 1.

例如：∫₀¹ x·eˣ dx，取 u = x，dv = eˣ dx，则 du = dx，v = eˣ。于是 [x·eˣ]₀¹ − ∫₀¹ eˣ dx = e − (e − 1) = 1。

5. Area Between a Curve and the x-Axis | 曲线与x轴围成的面积

The area between a curve y = f(x) and the x-axis from x = a to x = b is given by the definite integral of |f(x)|. If f(x) is always positive on [a, b], the area is simply ∫ₐᵇ f(x) dx.

曲线 y = f(x) 与 x 轴在 [a, b] 间围成的面积由定积分 ∫ₐᵇ |f(x)| dx 给出。若 f(x) 在 [a, b] 上恒正，则面积就是 ∫ₐᵇ f(x) dx。

When a curve crosses the x-axis, the integral over the whole interval can be zero even though the total area is not. You must split the region at each root and add the absolute values of the separate integrals.

当曲线穿过 x 轴时，整个区间上的定积分可能为 0，但总面积并不为 0。必须在每个零点处拆分区域，并将各段积分的绝对值相加。

Total area = ∫ₐᶜ f(x) dx + |∫꜀ᵈ f(x) dx| + ∫ᵈᵇ f(x) dx

For example, y = sin x on [0, 2π] has total area 4, while ∫₀²π sin x dx = 0.

例如，y = sin x 在 [0, 2π] 上总面积为 4，而 ∫₀²π sin x dx = 0。

6. Area Between Two Curves | 两条曲线之间的面积

To find the area enclosed by two curves y = f(x) and y = g(x), first determine their intersection points. On any interval where f(x) ≥ g(x), the area is ∫ (f(x) − g(x)) dx.

求两条曲线 y = f(x) 与 y = g(x) 围成的面积，先求交点。在 f(x) ≥ g(x) 的区间上，面积为 ∫ (f(x) − g(x)) dx。

Area = ∫ₐᵇ |f(x) − g(x)| dx

In exam problems, the two curves usually intersect at accessible points, giving natural limits of integration. If they cross multiple times, split the integral at each intersection.

考试中两条曲线通常在有意义的点相交，形成自然的积分限。如果交点有多个，需要在每个交点处拆分积分。

Always sketch the curves if possible—this helps determine which function is above the other and avoids sign mistakes.

如果可能，先画出曲线草图，以判断哪个函数在上方，避免符号错误。

7. Volume of Revolution | 旋转体体积

When the region under y = f(x) between x = a and x = b is rotated through 360° about the x-axis, the volume of the resulting solid is given by:

当 y = f(x) 与 x 轴之间从 x = a 到 x = b 的区域绕 x 轴旋转 360°，所得旋转体的体积为：

V = π ∫ₐᵇ [f(x)]² dx

If the rotation is about the y-axis and x is a function of y, the volume is V = π ∫꜀ᵈ [g(y)]² dy, where the limits c and d are y-values.

若绕 y 轴旋转且 x 是 y 的函数，则体积为 V = π ∫꜀ᵈ [g(y)]² dy，其中上下限 c 和 d 是 y 的取值。

Common pitfalls: forgetting the factor of π, or using the original function instead of its square. Always expand the square before integrating.

常见陷阱：漏乘 π，或者直接对被积函数本身积分而没有平方。务必先将平方展开再积分。

8. Mean Value of a Function | 函数的平均值

The average (mean) value of a continuous function f(x) over an interval [a, b] is defined as:

连续函数 f(x) 在区间 [a, b] 上的平均值定义为：

Average = (1/(b−a)) ∫ₐᵇ f(x) dx

This is a direct application of integration, and it is commonly used in physics contexts such as average velocity or average temperature.

这是积分的直接应用，常用于物理情境，例如平均速度或平均温度。

In Edexcel questions, the interval is often given explicitly. Be careful with fractions and units; the mean value is a number with the same units as f(x).

在 Edexcel 考题中，区间常常明确给出。注意分数和单位：平均值是一个数值，其单位与 f(x) 相同。

9. The Trapezium Rule | 梯形法则

When an indefinite integral cannot be found easily, numerical methods approximate the definite integral. The trapezium rule divides the interval into n strips of equal width h and estimates the area as a series of trapeziums.

当原函数不易求出时，可以用数值方法近似计算定积分。梯形法则把区间分成 n 个等宽为 h 的条带，并用一系列梯形近似面积。

∫ₐᵇ f(x) dx ≈ (h/2)[y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]

Here h = (b−a)/n, and yᵢ = f(a + i·h). The rule gives an overestimate if the curve is concave up, and an underestimate if it is concave down.

其中 h = (b−a)/n，yᵢ = f(a + i·h)。若曲线下凹（concave up），则此法则会高估；若上凸（concave down），则低估。

In exams, you may be asked to use a given number of strips. Write down the y-values in a table and substitute carefully.

考试中可能要求使用给定条数。先在表格中列出 y 值，再仔细代入公式。

10. Kinematics: Displacement, Velocity and Distance | 运动学：位移、速度和路程

Definite integrals appear naturally in mechanics. Given a velocity function v(t), the displacement between t = t₁ and t = t₂ is:

运动学中自然会出现定积分。已知速度函数 v(t)，则从 t = t₁ 到 t = t₂ 的位移为：

Displacement = ∫ₜ₁ᵗ² v(t) dt

However, total distance travelled is the integral of speed, i.e. |v(t)|:

但总路程是速度大小（速率）的积分，即 |v(t)|：

Distance = ∫ₜ₁ᵗ² |v(t)| dt

If v(t) changes sign, you must split the interval at the turning points. Displacement can be positive or negative, whereas distance is always non-negative.

若 v(t) 改变符号，必须在转向点处拆分区间。位移可正可负，而路程总是非负。

For example, if v(t) = t² − 3t + 2, the particle reverses direction at t = 1 and t = 2. To find the total distance from t = 0 to t = 3, integrate |v(t)| in three pieces.

例如，若 v(t) = t² − 3t + 2，质点在 t = 1 和 t = 2 处反向。要求从 t = 0 到 t = 3 的总路程，需分三段对 |v(t)| 积分。

11. Common Difficulties and Exam Tips | 常见难点与考试技巧

Students often make mistakes when dealing with negative areas, limits after substitution, or constants of integration. Here are some tips for definite integrals.

学生在处理负面积、换元后的上下限或积分常数时经常出错。以下是一些关于定积分的建议。

Always check whether the function is positive on the interval before finding area.

求面积前先检查函数在区间上是否恒为正。
When using substitution, do not ignore the change of limits; otherwise the result will be wrong.

使用换元法时，不要忽略上下限的变换，否则结果必然错误。
For integration by parts, write the evaluated term [u·v]ₐᵇ clearly—many errors occur here.

对于分部积分，要清晰写出求值项 [u·v]ₐᵇ，许多错误都发生在这里。
If a question asks for an area, give your answer as a positive number.

若题目要求面积，答案必须是正数。
Remember that π is outside the volume integral; do not square it accidentally.

记住 π 在体积积分号外面，不要无意中对其平方。
Use your calculator to verify final numerical answers in multiple-choice or defined questions.

在选择题或数字型题目中，可以用计算器验证最终答案。

12. Summary | 总结

Definite integration is not just a mechanical process but a versatile tool for solving geometric and physical problems. Mastering the fundamental theorem, properties, substitution, and parts lays the foundation; applying these techniques to area, volume, mean value, and kinematics builds exam confidence.

定积分不仅是一种机械运算，更是解决几何与物理问题的多功能工具。掌握微积分基本定理、性质、换元法与分部积分法打下基础；将这些技巧应用于面积、体积、平均值和运动学，能增强考试的信心。

The key to success is practice: work through past Edexcel questions, draw diagrams where needed, and always express the meaning behind the notation.

成功的关键在于练习：多做 Edexcel 历年真题，必要的时候画图，并且始终理解记号背后的含义。

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Definite Integration: Methods and Applications | 定积分的计算方法与应用