A-Level Mathematics: Finding the Area Between a Curve and a Line | A-Level 数学：曲线与直线之间面积的求法

📚 A-Level Mathematics: Finding the Area Between a Curve and a Line | A-Level 数学：曲线与直线之间面积的求法

In A-Level Mathematics, one of the most frequently tested applications of definite integration is calculating the area enclosed between a curve and a straight line. This topic combines solving simultaneous equations, identifying inequalities, and evaluating definite integrals, making it a rich source of exam questions.

在 A-Level 数学中，定积分最常考的用途之一就是计算曲线与直线之间所围成的面积。这一考点同时涉及联立方程求解、判断大小关系和计算定积分，因此也是考试题目的重要来源。

1. The Core Idea | 核心理念

The area between two functions y = f(x) and y = g(x) over an interval [a, b] is found by integrating the vertical distance between them. If f(x) is always greater than or equal to g(x) on the whole interval, then the required area is

两个函数 y = f(x) 与 y = g(x) 在区间 [a, b] 上围成的面积，可以通过对它们之间的竖直距离求积分得到。如果 f(x) 在整个区间内始终大于或等于 g(x)，那么所求面积就是

Area = ∫ₐᵇ [f(x) − g(x)] dx

If the line happens to lie above the curve, simply reverse the order of subtraction. In general terms, one can write Area = ∫ₐᵇ |f(x) − g(x)| dx, but in practice we always determine which function is upper before integrating.

如果直线恰好位于曲线上方，则把相减的次序反过来即可。一般地，可以写成 Area = ∫ₐᵇ |f(x) − g(x)| dx，但在实际操作中，我们总是先判断哪个函数在上方，再积分。

2. Why Integrate the Difference? | 为什么对“差”求积分？

Imagine slicing the required region into many thin vertical strips, each of width dx. The height of a strip is the vertical distance between the two graphs, namely (upper function − lower function). The area of one strip is roughly [upper − lower] × dx, and summing all strips from x = a to x = b gives the definite integral of the difference.

可以把所求区域想象成许多条宽为 dx 的竖直细条。每条细条的高度等于两条图像之间的竖直距离，也就是（上方函数 − 下方函数）。一条细条的面积近似为 [上方函数 − 下方函数] × dx，把从 x = a 到 x = b 的所有细条加起来，就得到这两个函数之差的定积分。

This is a direct extension of the familiar idea that the area under a single curve y = f(x) is ∫ₐᵇ f(x) dx. Here, instead of measuring height from the x-axis, we measure height from the lower graph to the upper graph.

这是“单条曲线 y = f(x) 下方的面积为 ∫ₐᵇ f(x) dx”这一常见概念的直接延伸。只不过这里不是从 x 轴量起，而是从下方图像量到上方图像。

3. Step 1 – Find the Points of Intersection | 第一步：求交点

The lower limit a and upper limit b are the x-coordinates of the points where the curve and the line meet. To find them, set the curve equation equal to the line equation:

下限 a 与上限 b 是曲线与直线交点处的横坐标。求法是把曲线方程与直线方程联立：

f(x) = g(x)

Solve this equation for x. Every real solution corresponds to an intersection point. For example, if the curve is y = x² and the line is y = x + 2, then x² = x + 2 leads to x² − x − 2 = 0, which factors as (x − 2)(x + 1) = 0, giving x = −1 and x = 2.

解出 x 即可。每个实数解都对应一个交点。例如，若曲线为 y = x²，直线为 y = x + 2，则 x² = x + 2 可化为 x² − x − 2 = 0，分解因式得 (x − 2)(x + 1) = 0，所以 x = −1 与 x = 2。

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