Area Between Two Curves | 两函数间面积的计算方法

📚 Area Between Two Curves | 两函数间面积的计算方法

The area enclosed between two curves is a fundamental application of definite integrals in IB Mathematics. This technique allows us to measure the region bounded by the graphs of two functions, often within a given interval or between their intersection points.

两函数所夹区域的面积是IB数学中定积分的重要应用。这一方法能够帮助我们度量由两个函数图像所围成的区域面积，通常是在给定区间内或两个函数交点之间。

1. Understanding the Basic Concept | 理解面积的基本概念

When we talk about the area between two curves, we mean the total region that lies between the graphs of two functions, y = f(x) and y = g(x). This area is measured by summing up infinitely thin vertical strips of width dx and height equal to the difference between the two functions.

当我们讨论两函数之间的面积时，我们指的是函数 y = f(x) 和 y = g(x) 图像之间所夹的全部区域。这个面积可以通过将无数个宽度为 dx、高度等于两函数之差的竖直细条求和来得到。

The height of each strip is |f(x) − g(x)|.
The width of each strip is dx.
Integration sums the areas of all strips over the desired interval.

每个细条的高度为 |f(x) − g(x)|。
每个细条的宽度为 dx。
通过积分将所有细条的面积在指定区间内求和。

2. Why Integration? | 为什么是积分？

Integration is the process of accumulating quantities. Since the heights of the strips vary with x, we cannot simply multiply base by height. Integration provides a rigorous way to sum an infinite number of infinitesimal areas.

积分是累积量的过程。由于细条的高度随 x 变化，我们不能简单地进行底乘高。积分为我们提供了对无穷多个无穷小面积求和的严谨方法。

The definite integral of a positive function gives the area under the curve. For two functions, the area between them is the integral of the vertical difference.

定积分对正函数求值可得到曲线下的面积。对于两个函数，它们之间的面积等于其竖直之差的积分。

3. The Core Formula: Top minus Bottom | 核心公式：上减下

If f(x) ≥ g(x) for all x in [a, b], the area between the two curves is given by the integral of f(x) − g(x) from a to b.

如果在 [a, b] 上恒有 f(x) ≥ g(x)，则两曲线之间的面积等于从 a 到 b 对 f(x) − g(x) 的积分。

A = ∫_a^b [f(x) − g(x)] dx

This formula is intuitive: at each x, the vertical distance between the curves is f(x) − g(x), and integrating these distances over the interval gives the total area.

这个公式非常直观：在每个 x 处，曲线之间的竖直距离为 f(x) − g(x)，在区间上对这些距离积分便得到总面积。

4. Finding Intersection Points: Determining Limits | 确定交点：求积分限

When the interval is not specified, the limits of integration are the x-coordinates of the intersection points of the two curves. These are found by solving f(x) = g(x).

当题目未指定区间时，积分上下限为两曲线交点的 x 坐标。通过解方程 f(x) = g(x) 即可求得这些点。

For example, to find the area bounded by y = x² and y = x + 2, we solve x² = x + 2, which gives x² − x − 2 = 0, so x = −1 and x = 2.

例如，求 y = x² 和 y = x + 2 所围成的面积，我们解 x² = x + 2，得到 x² − x − 2 = 0，所以 x = −1 和 x = 2。

Always check which function is on top within the interval, because the upper function might change across different subintervals.

始终检查区间内哪个函数在上方，因为上下关系可能在不同子区间内发生改变。

5. When Functions Cross: Piecewise Handling | 当函数交叉时：分段处理

If the two curves intersect within the interval of interest, the “top” function changes. We must split the integral at each intersection point and use the absolute difference to ensure a positive area contribution.

如果两条曲线在所讨论的区间内相交，那么“上方”函数会发生变化。我们必须将积分在每个交点处分段，并使用绝对值差以确保面积贡献为正。

The general formula becomes:

一般公式变为：

A = ∫_a^c |f(x) − g(x)| dx + ∫_c^b |f(x) − g(x)| dx

where c is an intersection point inside (a, b). In practice, we determine the sign of f(x) − g(x) in each subinterval and integrate accordingly.

其中 c 是 (a, b) 内的交点。实际操作中，我们判断每个子区间内 f(x) − g(x) 的符号，然后分别积分。

6. Integrating with Respect to y: Right minus Left | 关于 y 轴积分：右减左

Some regions are more conveniently described as functions of y, especially when the curves are given as x = h(y) and x = k(y). In such cases, the area is the integral of the horizontal distance (right curve minus left curve) with respect to y.

有些区域更适合用 y 的函数来描述，特别是当曲线以 x = h(y) 和 x = k(y) 形式给出时。此时，面积等于水平距离（右曲线减左曲线）对 y 的积分。

A = ∫_c^d [h(y) − k(y)] dy

Here c and d are the y-coordinates of the intersection points (or the boundaries of the region). This method is particularly useful for functions with multiple x-values for a single y-value.

这里的 c 和 d 是交点的 y 坐标（或区域的边界）。当某个 y 值对应多个 x 值时，这种方法特别有用。

7. The Role of Absolute Value | 绝对值的作用

Since area is always positive, we must ensure that the integrand is nonnegative. The absolute value |f(x) − g(x)| guarantees this. However, in many simple problems, we know the relative order and can avoid absolute values by properly subtracting the lower function from the upper function.

由于面积总是正的，我们必须确保被积函数非负。绝对值 |f(x) − g(x)| 可以保证这一点。然而，在许多简单问题中，我们知道函数的相对位置，可以通过用上方函数减去下方函数来避免使用绝对值。

Using the absolute value in the integral gives a compact but often less practical form, because it requires piecewise evaluation. Drawing the graphs first is strongly recommended.

在积分中使用绝对值虽然形式简洁，但往往不太实用，因为它需要分段求值。强烈建议先画出图像。

8. Area Between a Curve and the x-axis | 曲线与 x 轴之间的面积

When one function is the x-axis, i.e. g(x) = 0, the area reduces to the integral of |f(x)|. This special case is important because it highlights that the signed area under the x-axis must be taken as positive when computing physical area.

当其中一个函数为 x 轴，即 g(x) = 0 时，面积就简化为对 |f(x)| 的积分。这一特殊情况非常重要，因为它说明在计算实际面积时，x 轴下方的有符号面积必须取为正。

For example, the area between y = x³ − x and the x-axis from x = −1 to x = 1 requires splitting at x = 0, because x³ − x is positive on (−1, 0) and negative on (0, 1).

例如，计算 y = x³ − x 与 x 轴在 x = −1 到 x = 1 之间的面积时，需要在 x = 0 处分段，因为 x³ − x 在 (−1, 0) 上为正，在 (0, 1) 上为负。

9. Common Mistakes and Pitfalls | 常见错误与陷阱

Students often make mistakes when:

学生常在以下情况出错：

Using the wrong order of subtraction if the curves cross.
Forgetting to find intersection points and using arbitrary limits.
Applying the same expression over intervals where the upper function changes.
Confusing area with signed area when the curve goes below the x-axis.

当曲线交叉时使用了错误的相减顺序。
忘记求交点而随意使用积分限。
在上方函数变化的区间内使用同一个表达式。
当曲线低于 x 轴时，混淆有符号面积与实际面积。

A reliable strategy is to sketch the graphs, mark all intersections, and label which function is above in each subinterval.

一个可靠的策略是画出图像，标记所有交点，并在每个子区间内标出哪个函数在上方。

10. Step-by-Step Summary and Worked Example | 解题步骤总结与例题演练

Follow these steps for any “area between two curves” problem:

对于任何“两曲线间面积”问题，请遵循以下步骤：

Sketch or describe the curves.
Find all intersection points by solving f(x) = g(x).
Determine the interval [a, b] (either given or between intersections).
Identify the upper function on each subinterval.
Set up the integral of (upper − lower) with respect to x (or right − left with respect to y).
Evaluate the integral and interpret the result.

画出或描述曲线。
通过解 f(x) = g(x) 找到所有交点。
确定区间 [a, b]（给定或由交点决定）。
在每个子区间内确定上方函数。
建立对 x 的（上方 − 下方）积分，或对 y 的（右 − 左）积分。
计算积分并解释结果。

Worked example: Find the area enclosed by y = x² and y = 2x − x².

例题：求 y = x² 和 y = 2x − x² 所围成的面积。

First, solve x² = 2x − x² ⟹ 2x² − 2x = 0 ⟹ x = 0 or x = 1. On (0, 1), 2x − x² − x² = 2x − 2x² = 2x(1 − x) > 0, so the upper function is y = 2x − x².

首先，解 x² = 2x − x² ⟹ 2x² − 2x = 0 ⟹ x = 0 或 x = 1。在 (0, 1) 上，2x − x² − x² = 2x − 2x² = 2x(1 − x) > 0，所以上方函数为 y = 2x − x²。

A = ∫₀¹ [(2x − x²) − x²] dx = ∫₀¹ (2x − 2x²) dx = [x² − (2/3)x³]₀¹ = 1 − 2/3 = 1/3

The area is 1/3 square units.

面积为 1/3 平方单位。

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Area Between Two Curves | 两函数间面积的计算方法