📚 A-Level Mathematics: Finding Area Under Curves Using Definite Integrals | A-Level 数学：用定积分求曲线面积
The definite integral is one of the most powerful tools in A-Level mathematics. It allows us to calculate the exact area enclosed by a curve, the x-axis, and vertical lines, as well as the area between two curves. This article will guide you through the core concepts, standard methods, and common pitfalls, with clear worked examples aligned to the examination style.
定积分是 A-Level 数学中最强大的工具之一。它使我们能够精确计算由曲线、x 轴和垂直线围成的面积，以及两条曲线之间的面积。本文将指导你掌握核心概念、标准方法和常见误区，并提供与考试风格一致的清晰例题。
1. Understanding the Definite Integral and Area | 理解定积分与面积
The definite integral of a function f(x) from a to b, written as ∫ₐᵇ f(x) dx, represents the signed area between the graph of f(x) and the x-axis over the interval [a, b]. A positive value indicates area above the x-axis, while a negative value indicates area below it.
函数 f(x) 从 a 到 b 的定积分，记作 ∫ₐᵇ f(x) dx，表示 f(x) 的图像与 x 轴在区间 [a, b] 上有符号的面积。正值表示 x 轴上方的面积，负值表示 x 轴下方的面积。
The fundamental theorem of calculus links this area to antiderivatives: if F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) – F(a). You will use this directly in almost every area problem.
微积分基本定理将面积与原函数联系起来：若 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) – F(a)。在几乎所有的面积问题中，你都会直接使用这一关系。
2. Area Above the x-Axis | x 轴上方的面积
When the curve lies entirely above the x-axis on [a, b], the integral itself gives the required area. No modification is needed; simply evaluate the definite integral.
当曲线在 [a, b] 上完全位于 x 轴上方时，定积分本身即为所求面积，无需修改，直接计算定积分即可。
Example: Find the area enclosed by y = x², the x-axis, and the lines x = 0 and x = 2.
例：求 y = x²、x 轴以及直线 x = 0 和 x = 2 所围成的面积。
A = ∫₀² x² dx = [x³/3]₀² = 8/3 – 0 = 8/3
Hence the area is 8/3 square units. Notice that the antiderivative x³/3 is evaluated at the upper limit first, then the lower limit.
因此面积为 8/3 个平方单位。注意先代入上限，再代入下限计算原函数 x³/3。
3. Area Below the x-Axis | x 轴下方的面积
If the curve lies entirely below the x-axis, the definite integral returns a negative value. The physical area is the absolute value of that integral.
若曲线完全位于 x 轴下方，定积分得到负值，实际面积是该积分的绝对值。
Example: Find the area enclosed by y = x³ – 4x, the x-axis, and the lines x = 0 and x = 2.
例：求 y = x³ – 4x、x 轴以及直线 x = 0 和 x = 2 所围成的面积。
∫₀² (x³ – 4x) dx = [x⁴/4 – 2x²]₀² = (4 – 8) – 0 = -4
Since the curve is below the x-axis on this interval, the area is 4 square units.
由于曲线在该区间上位于 x 轴下方，所以面积为 4 个平方单位。
4. Area Crossing the x-Axis | 曲线跨越 x 轴的情形
When a curve crosses the x-axis within the integration interval, the integral cancels positive and negative contributions. You must split the interval at the x-intercepts and take the absolute value of each part before adding.
当曲线在积分区间内穿过 x 轴时，积分会抵消正负部分。你必须在 x 轴截点处分割区间，先对各部分取绝对值再相加。
For example, consider y = x³ – 4x on [-2, 2]. It crosses the x-axis at x = -2, 0, 2. The total enclosed area is given by:
例如，考虑 y = x³ – 4x 在 [-2, 2] 上。曲线在 x = -2, 0, 2 处穿过 x 轴。所围成的总面积为：
|∫₋₂⁰ (x³ – 4x) dx| + |∫₀² (x³ – 4x) dx| = 4 + 4 = 8
Always check for roots inside the interval before evaluating the area. The single integral ∫₋₂² (x³ – 4x) dx would incorrectly give 0.
在求面积前一定要检查区间内是否存在根。若直接计算 ∫₋₂² (x³ – 4x) dx 会错误地得到 0。
5. Area Between Two Curves | 两条曲线之间的面积
The area between two curves y = f(x) (upper) and y = g(x) (lower) on [a, b] is given by the integral of the difference (top minus bottom):
两条曲线 y = f(x)（上方）和 y = g(x)（下方）在 [a, b] 上围成的面积为两者之差（上减下）的积分：
A = ∫ₐᵇ [f(x) – g(x)] dx
This formula automatically gives a positive result as long as f(x) ≥ g(x) throughout the interval. If the curves cross, split the interval at the crossing points.
只要在区间内始终有 f(x) ≥ g(x)，该公式自动给出正值。若曲线相交，则在交点处分割区间。
Example: Find the area between y = x and y = x². First find intersections: x = x² gives x = 0 or x = 1. On [0, 1], y = x is the upper curve.
例：求 y = x 与 y = x² 之间的面积。先求交点：x = x² 得 x = 0 或 x = 1。在 [0, 1] 上，y = x 是上方曲线。
A = ∫₀¹ (x – x²) dx = [x²/2 – x³/3]₀¹ = 1/2 – 1/3 = 1/6
Hence the area is 1/6 square units.
因此面积为 1/6 个平方单位。
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