Directed Areas | 有向面积

📚 Directed Areas | 有向面积

In A-Level mathematics, the definite integral of a function can be interpreted as a directed area. Unlike the purely geometric area, the directed area carries a sign that depends on the position of the curve relative to the x-axis. Understanding this distinction is essential for solving problems involving integration and area calculation.

在 A-Level 数学中，函数定积分可以解释为有向面积。与纯几何面积不同，有向面积带有符号，该符号取决于曲线相对于 x 轴的位置。理解这种区别对于解决涉及积分和面积计算的问题至关重要。

1. What Is a Directed Area? | 什么是有向面积？

A directed area is an area whose value can be positive, negative, or zero, depending on the orientation of the region relative to the coordinate axes. When we evaluate the integral of a function over an interval, the result represents the net signed area between the curve and the x-axis.

有向面积是一种面积，其值可以为正、负或零，具体取决于区域相对于坐标轴的取向。当我们计算一个函数在某个区间上的积分时，结果表示曲线与 x 轴之间的净带符号面积。

Key idea: The sign of the directed area is determined by whether the curve lies above (positive contribution) or below (negative contribution) the x-axis.

关键思想： 有向面积的符号由曲线位于 x 轴上方（正贡献）还是下方（负贡献）决定。

2. Definite Integral as Directed Area | 定积分作为有向面积

For a continuous function f(x) on the interval [a, b], the definite integral from a to b is the directed area between y = f(x) and the x-axis. Each infinitesimal strip of width dx and height f(x) contributes f(x) dx, which may be positive or negative.

对于区间 [a, b] 上的连续函数 f(x)，从 a 到 b 的定积分是 y = f(x) 与 x 轴之间的有向面积。每个宽度为 dx、高度为 f(x) 的无穷小条带贡献 f(x) dx，该值可能为正或为负。

∫ₐᵇ f(x) dx = directed area from x = a to x = b

∫ₐᵇ f(x) dx = 从 x = a 到 x = b 的有向面积

3. Positive and Negative Regions | 正区域与负区域

If f(x) > 0 throughout an interval, the directed area is positive. If f(x) < 0 throughout an interval, the directed area is negative. If the curve crosses the x-axis, the contributions from different sub-intervals are added algebraically.

如果 f(x) 在整个区间内大于 0，则有向面积为正。如果 f(x) 在整个区间内小于 0，则有向面积为负。如果曲线穿过 x 轴，不同子区间的贡献将按代数相加。

Above the x-axis: f(x) > 0 → integral gives positive area.
Below the x-axis: f(x) < 0 → integral gives negative area.

x 轴上方： f(x) > 0 → 积分给出正面积。
x 轴下方： f(x) < 0 → 积分给出负面积。

4. The Fundamental Trap: Geometric Area vs Directed Area | 基本陷阱：几何面积与有向面积

Examination questions often ask for the “area” of a region. If the curve dips below the x-axis, simply integrating from the lower limit to the upper limit will give a net directed area, which may be too small because negative contributions cancel positive ones. Always consider whether the question wants directed area or total geometric area.

考试题目经常要求某个区域的“面积”。如果曲线下降到 x 轴以下，简单地从下限积分到上限会得到净有向面积，该值可能太小，因为负贡献抵消了正贡献。始终考虑题目要求的是有向面积还是总几何面积。

Total geometric area = ∫ₐᵇ |f(x)| dx ≠ ∫ₐᵇ f(x) dx

总几何面积 = ∫ₐᵇ |f(x)| dx ≠ ∫ₐᵇ f(x) dx

5. Example: Positive Directed Area | 示例：正有向面积

Find the directed area bounded by y = x², the x-axis, and the lines x = 1 and x = 3.

求由 y = x²、x 轴以及直线 x = 1 和 x = 3 围成的有向面积。

∫₁³ x² dx = [x³/3]₁³ = 27/3 − 1/3 = 26/3

Since x² is always positive on [1, 3], the directed area equals the geometric area, which is 26/3 square units.

由于 x² 在 [1, 3] 上始终为正，有向面积等于几何面积，即 26/3 平方单位。

6. Example: Negative Directed Area | 示例：负有向面积

Evaluate the directed area between y = −x and the x-axis from x = 0 to x = 2.

计算 y = −x 与 x 轴之间从 x = 0 到 x = 2 的有向面积。

∫₀² (−x) dx = [−x²/2]₀² = −4/2 − 0 = −2

The integral is negative because the curve lies entirely below the x-axis. The geometric area is actually 2 square units, but the directed area is −2.

积分值为负，因为曲线完全位于 x 轴下方。几何面积实际上为 2 平方单位，但有向面积为 −2。

7. Splitting the Interval at Roots | 在根处分割区间

To find the total geometric area between a curve and the x-axis, first solve f(x) = 0 to find the roots within the interval. Then integrate f(x) over each sub-interval separately, take the absolute value of each result, and sum them.

为了求曲线与 x 轴之间的总几何面积，首先解 f(x) = 0 以找到区间内的根。然后分别对每个子区间积分 f(x)，取每个结果的绝对值，并将它们相加。

Procedure:

Find roots of f(x) in [a, b].
Integrate f(x) over each sub-interval.
Add the absolute values of the integrals.

步骤：

找到 f(x) 在 [a, b] 内的根。
在每个子区间上对 f(x) 积分。
将各积分的绝对值相加。

8. Worked Example: Cubic with a Negative Dip | 示例：带有负下凹的三次函数

Find the total area enclosed by y = x(x − 1)(x − 2), the x-axis, and the vertical lines x = 0 and x = 2.

求由 y = x(x − 1)(x − 2)、x 轴以及垂直线 x = 0 和 x = 2 所围成的总面积。

First, expand: y = x³ − 3x² + 2x. Roots are x = 0, 1, 2. On (0, 1), the curve is positive; on (1, 2), it is negative.

首先展开：y = x³ − 3x² + 2x。根为 x = 0、1、2。在 (0, 1) 上，曲线为正；在 (1, 2) 上，曲线为负。

∫₀¹ (x³ − 3x² + 2x) dx = [x⁴/4 − x³ + x²]₀¹ = 1/4 − 1 + 1 = 1/4

∫₁² (x³ − 3x² + 2x) dx = [x⁴/4 − x³ + x²]₁² = (4 − 8 + 4) − (1/4 − 1 + 1) = 0 − 1/4 = −1/4

Total geometric area = |1/4| + |−1/4| = 1/2. The directed area is 1/4 + (−1/4) = 0.

总几何面积 = |1/4| + |−1/4| = 1/2。有向面积为 1/4 + (−1/4) = 0。

9. Area Between Two Curves | 两曲线之间的面积

For the area between two curves y = f(x) and y = g(x), we integrate the absolute difference |f(x) − g(x)| over the interval. However, when the curves intersect, we must split the integral at the intersection points because the order of the curves may change.

对于两条曲线 y = f(x) 和 y = g(x) 之间的面积，我们在区间上积分差值绝对值 |f(x) − g(x)|。然而，当曲线相交时，我们必须在交点处分割积分，因为曲线的上下顺序可能改变。

Area = ∫ₐᵇ |f(x) − g(x)| dx

面积 = ∫ₐᵇ |f(x) − g(x)| dx

10. Parameterised Curves: Directed Area | 参数曲线：有向面积

For a curve defined parametrically by x = x(t), y = y(t), the directed area swept out between the curve and the x-axis from t = t₁ to t = t₂ is given by the integral of y dx = y (dx/dt) dt. The sign depends on the direction of traversal.

对于由 x = x(t)、y = y(t) 定义的参数曲线，从 t = t₁ 到 t = t₂ 在曲线与 x 轴之间扫过的有向面积由 y dx = y (dx/dt) dt 的积分给出。符号取决于遍历方向。

Directed area = ∫ₜ₁ᵗ² y(t) x′(t) dt

有向面积 = ∫ₜ₁ᵗ² y(t) x′(t) dt

11. Common Mistakes and Exam Tips | 常见错误与考试提示

Mistake: Forgetting to split at roots when the curve crosses the x-axis.
Tip: Always sketch the graph if possible.
Mistake: Confusing directed area with geometrical area.
Tip: Read carefully: “area” usually means positive area; “integral” can be signed.
Mistake: Incorrectly applying limits when finding intersections.
Tip: Solve f(x) = g(x) to find limits for area between curves.

错误： 当曲线穿过 x 轴时忘记在根处分割区间。
提示： 尽可能先画草图。
错误： 混淆有向面积与几何面积。
提示： 仔细读题：“面积”通常指正面积，“积分”可能带符号。
错误： 求交点时错误应用积分上下限。
提示： 解 f(x) = g(x) 以求得两曲线之间面积的限。

12. Practice Quick Check | 快速练习

Try these two problems to consolidate the concept.

尝试以下两个问题来巩固概念。

Evaluate ∫₋₁¹ x³ dx. Is this the geometric area? Explain.
Find the total area bounded by y = sin x, the x-axis, from x = 0 to x = 2π.

计算 ∫₋₁¹ x³ dx。这是几何面积吗？解释原因。
求 y = sin x 与 x 轴从 x = 0 到 x = 2π 所围成的总面积。

Answers: 1) The integral equals 0 because x³ is an odd function; the directed areas cancel. The geometric area is ∫₋₁¹ |x³| dx = 1/2. 2) Since sin x is positive on (0, π) and negative on (π, 2π), split: the total area is 2 + 2 = 4.

答案： 1) 积分等于 0，因为 x³ 是奇函数；有向面积相互抵消。几何面积为 ∫₋₁¹ |x³| dx = 1/2。2) 因为 sin x 在 (0, π) 为正，在 (π, 2π) 为负，分割：总面积为 2 + 2 = 4。

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Directed Areas | 有向面积