Areas Under Curves | 曲线下方面积

📚 Areas Under Curves | 曲线下方面积

In A-Level Edexcel Mathematics, the topic ‘Areas under curves’ brings together definite integration and geometric interpretation. You need to be confident in using integration to find the area between a curve and the x-axis, handling regions below the x-axis correctly, and calculating the area enclosed between two different curves. This article explains the key ideas, common pitfalls, and exam-ready methods.

在 A-Level 爱德思数学中，“曲线下方面积”这一主题将定积分与几何意义结合起来。你需要熟练运用积分求曲线与 x 轴之间的面积，正确处理 x 轴下方的区域，并计算两条不同曲线所围成的面积。本文讲解核心概念、常见易错点和考试常用方法。

1. The Definite Integral as Area | 定积分与面积

The definite integral of a non-negative function f(x) from x = a to x = b gives the exact area between the curve y = f(x), the x-axis, and the vertical lines x = a and x = b. This only works directly when f(x) ≥ 0 throughout the interval [a, b].

非负函数 f(x) 从 x = a 到 x = b 的定积分给出曲线 y = f(x)、x 轴以及直线 x = a 和 x = b 之间的精确面积。只有当 f(x) 在整个区间 [a, b] 上始终大于等于 0 时，才能直接这样使用。

Area = ∫ₐᵇ f(x) dx

The notation ∫ₐᵇ means the integral from a to b. The process of finding an integral and substituting the limits is called evaluating the definite integral.

符号 ∫ₐᵇ 表示从 a 到 b 的积分。求积分并代入上下限的过程称为计算定积分。

The integrand f(x) is the expression being integrated.
dx indicates the variable of integration.
The limits a and b are the lower and upper bounds.

被积函数 f(x) 是被积分的表达式。dx 表示积分变量。下限 a 和上限 b 是积分的边界。

2. Signed Area and the x-Axis | 有向面积与 x 轴

When a function takes negative values, the definite integral gives a signed area, not the actual geometric area. Parts above the x-axis contribute positive values, while parts below the x-axis contribute negative values. If the curve crosses the x-axis within the interval, simply integrating from a to b may cancel out some area.

当函数取负值时，定积分给出的是有向面积，而不是实际的几何面积。x 轴上方的部分贡献正值，x 轴下方的部分贡献负值。如果曲线在区间内穿过 x 轴，直接从 a 到 b 积分可能会导致部分面积相互抵消。

For example, if f(x) = x from -2 to 2, the integral from -2 to 2 is 0 because the positive area from 0 to 2 and the negative area from -2 to 0 cancel exactly. The true geometric area is 4 square units, not 0.

例如，若 f(x) = x 在 -2 到 2 之间，从 -2 到 2 的积分为 0，因为 0 到 2 的正面积和 -2 到 0 的负面积正好抵消。真实的几何面积是 4 平方单位，而不是 0。

Therefore, you must always check whether the curve crosses the x-axis before applying the simple formula Area = ∫ₐᵇ f(x) dx.

因此，在直接使用公式 Area = ∫ₐᵇ f(x) dx 之前，你必须始终检查曲线是否穿过 x 轴。

3. Areas Below the x-Axis | x 轴下方的面积

To find the actual area of a region below the x-axis, take the absolute value of the definite integral. If f(x) ≤ 0 on [a, b], then the area is given by:

要求 x 轴下方区域的实际面积，应取定积分的绝对值。如果 f(x) 在 [a, b] 上小于等于 0，则面积为：

Area = | ∫ₐᵇ f(x) dx |

Alternatively, you can integrate the negative of the function, or write the integral with a minus sign in front. This ensures the final area is positive.

另一种方法是积分函数的相反数，或者在积分前加上负号。这样可以保证最终面积为正值。

For f(x) = x² – 4 from x = 0 to x = 2, the curve lies below the x-axis. The integral ∫₀² (x² – 4) dx equals -16/3, so the actual area is 16/3 square units.

对于 f(x) = x² – 4，从 x = 0 到 x = 2，曲线位于 x 轴下方。积分 ∫₀² (x² – 4) dx 等于 -16/3，因此实际面积是 16/3 平方单位。

In exam solutions, always state that you are taking the modulus or changing the sign because the region is below the x-axis.

在考试解答中，务必说明你正在取绝对值或改变符号，因为该区域位于 x 轴下方。

4. Area Between a Curve and the x-Axis in Intervals | 曲线与 x 轴在区间内围成的面积

When the curve crosses the x-axis one or more times within the interval, split the interval at each root. Integrate separately over each sub-interval, take the modulus of each integral, and then add the results together.

当曲线在区间内一次或多次穿过 x 轴时，需在每个根处拆分区间。分别对每个子区间积分，对每个积分取绝对值，然后将结果相加。

For a curve y = f(x) that crosses the x-axis at x = c, where a < c < b, the total area is:

对于曲线 y = f(x) 在 x = c 处穿过 x 轴，其中 a < c < b，总面积为：

Area = | ∫ₐᶜ f(x) dx | + | ∫ᶜᵇ f(x) dx |

This method prevents positive and negative contributions from cancelling. You should always sketch the curve or find the sign of f(x) between the roots.

这种方法可以防止正值和负值贡献相互抵消。你应该始终画出曲线草图，或判断 f(x) 在各根之间的符号。

Find the roots of f(x) = 0 in the interval.
Split the integral at each root.
Evaluate each part separately and take absolute values.

求出区间内 f(x) = 0 的根。在每个根处拆分积分。分别计算每一部分并取绝对值。

5. Area Between Two Curves | 两曲线之间的面积

To find the area enclosed between two curves y = f(x) and y = g(x) from x = a to x = b, integrate the difference between the upper and lower functions. The formula is:

要求两条曲线 y = f(x) 和 y = g(x) 在 x = a 到 x = b 之间围成的面积，应对上方函数与下方函数的差进行积分。公式为：

Area = ∫ₐᵇ (f(x) – g(x)) dx

Here f(x) is the upper curve and g(x) is the lower curve on the interval. If the curves intersect within the interval, split the integral at the intersection points because the upper and lower functions may swap.

这里 f(x) 是区间上的上方曲线，g(x) 是下方曲线。如果两曲线在区间内相交，则应在交点处拆分积分，因为上下方函数可能会互换。

For example, the area between y = x² and y = x + 2 is found by first locating their intersection at x = -1 and x = 2. On this interval, y = x + 2 is above y = x², so:

例如，y = x² 与 y = x + 2 之间的面积可先求出它们的交点为 x = -1 和 x = 2。在该区间上，y = x + 2 位于 y = x² 的上方，因此：

Area = ∫₋₁² (x + 2 – x²) dx

6. Using Symmetry to Simplify | 利用对称性简化

If a curve is symmetric about the y-axis, you can calculate the area on one side and then double it. For even functions such as y = x², y = cos x, or y = e^(x²), the area from -a to a is twice the area from 0 to a.

如果曲线关于 y 轴对称，你可以计算一侧的面积然后乘以 2。对于偶函数，如 y = x²、y = cos x 或 y = e^(x²)，从 -a 到 a 的面积等于从 0 到 a 的面积的两倍。

∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx

This shortcut is useful when the integration is easier from 0 to a, or when the equation contains even powers only. However, be careful if the function crosses the x-axis or if the region has both positive and negative parts.

当从 0 到 a 的积分更容易，或者方程只含有偶次幂时，这个技巧很有用。但要注意函数是否穿过 x 轴，或者区域是否既有正值部分又有负值部分。

For example, the area between y = x² and the x-axis from x = -2 to x = 2 is:

例如，y = x² 与 x 轴在 x = -2 到 x = 2 之间的面积为：

Area = 2 ∫₀² x² dx = 2 × 8/3 = 16/3

7. Parametric Curves: Area Formula | 参数曲线面积公式

When a curve is defined parametrically by x = x(t) and y = y(t), the area between the curve and the x-axis can be found using the formula:

当曲线由参数方程 x = x(t) 和 y = y(t) 定义时，曲线与 x 轴之间的面积可以用以下公式求出：

Area = ∫ y (dx/dt) dt

The limits are the parameter values corresponding to the desired x-values. This formula is derived from the chain rule and the standard area integral ∫ y dx.

积分限是与所需 x 值对应的参数值。该公式由链式法则和标准面积积分 ∫ y dx 推导而来。

For a parametric curve with x = t² and y = 2t, the area under the curve from t = 0 to t = 3 is:

对于参数曲线 x = t² 和 y = 2t，从 t = 0 到 t = 3 的曲线下方面积为：

Area = ∫₀³ (2t)(2t) dt = ∫₀³ 4t² dt = [4t³/3]₀³ = 36

Edexcel questions may ask you to find the area of a parametric region, so be ready to convert limits and use dx/dt correctly.

爱德思考题可能会要求你求参数区域面积，因此要准备好转换积分限并正确使用 dx/dt。

8. Common Mistakes and Exam Tips | 常见错误与考试技巧

One of the most common mistakes is forgetting to take the absolute value when the curve lies below the x-axis. Another is using the wrong upper and lower functions when finding the area between two curves.

最常见的错误之一是当曲线位于 x 轴下方时忘记取绝对值。另一个是在求两曲线之间面积时弄错上下方函数。

Always sketch the graph or determine the sign of f(x) before integrating.
Find intersection points when curves cross each other or the x-axis.
Check that your final area is positive.
Use exact values such as fractions and surds unless otherwise stated.
Remember to write the constant of integration only for indefinite integrals, not definite integrals.

在积分前始终画出图形草图或判断 f(x) 的符号。当曲线彼此相交或与 x 轴相交时，求出交点。检查最终面积是否为正值。除非另有说明，使用分数和根式等精确值。记住，只有不定积分才写积分常数，定积分不写。

In Edexcel exams, marks are often awarded for setting up the correct integral, finding limits, and evaluating accurately. Show all these steps clearly to earn full credit.

在爱德思考试中，正确建立积分式、求出积分限并准确计算往往都能得分。请清晰地展示这些步骤以获得满分。

9. Worked Example 1: Polynomial Curve | 例题1：多项式曲线

Find the total area enclosed between the curve y = x³ – 4x and the x-axis from x = 0 to x = 3.

求曲线 y = x³ – 4x 与 x 轴在 x = 0 到 x = 3 之间围成的总面积。

First, find where the curve crosses the x-axis by solving x³ – 4x = 0:

首先，通过解方程 x³ – 4x = 0 找出曲线与 x 轴的交点：

x(x² – 4) = 0 → x = 0, x = 2, x = -2

In the interval [0, 3], the relevant root is x = 2. From 0 to 2 the curve is below the x-axis, and from 2 to 3 it is above the x-axis.

在区间 [0, 3] 内，相关的根是 x = 2。从 0 到 2 曲线位于 x 轴下方，从 2 到 3 曲线位于 x 轴上方。

Area = | ∫₀² (x³ – 4x) dx | + | ∫₂³ (x³ – 4x) dx |

Integrate each part:

分别积分：

∫ (x³ – 4x) dx = x⁴/4 – 2x²

∫₀² = (16/4 – 8) – 0 = 4 – 8 = -4 → | -4 | = 4

∫₂³ = (81/4 – 18) – (4 – 8) = 81/4 – 18 + 4 = 81/4 – 14 = 25/4

Total Area = 4 + 25/4 = 41/4 square units

The final answer is 41/4 square units, not the signed value from a single integral from 0 to 3, which would be 9/4.

最终答案是 41/4 平方单位，而不是从 0 到 3 一次积分得到的有向值 9/4。

10. Worked Example 2: Area Between Curves | 例题2：两曲线之间的面积

Find the area enclosed between the curves y = x² and y = 2x + 3.

求曲线 y = x² 与 y = 2x + 3 所围成的面积。

First, find the intersection points by solving x² = 2x + 3:

首先，通过解方程 x² = 2x + 3 求交点：

x² – 2x – 3 = 0 → (x – 3)(x + 1) = 0 → x = -1, x = 3

On the interval [-1, 3], the line y = 2x + 3 is above the curve y = x². The area is therefore:

在区间 [-1, 3] 上，直线 y = 2x + 3 位于曲线 y = x² 上方。因此面积为：

Area = ∫₋₁³ (2x + 3 – x²) dx

Integrate term by term:

逐项积分：

∫ (2x + 3 – x²) dx = x² + 3x – x³/3

Substitute the limits:

代入上下限：

[x² + 3x – x³/3]₋₁³ = (9 + 9 – 27/3) – (1 – 3 + 1/3)

= (18 – 9) – (-5/3) = 9 + 5/3 = 32/3

The area enclosed between the two curves is 32/3 square units. This method is a standard Edexcel question type and often appears with quadratics and straight lines.

这两条曲线所围成的面积是 32/3 平方单位。这种方法是爱德思考试的标准题型，经常与二次函数和直线一起出现。

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Areas Under Curves | 曲线下方面积