A-Level Mathematics: Definite Integrals for Finding Area Under a Curve | A-Level数学：曲线下面积的定积分求法

📚 A-Level Mathematics: Definite Integrals for Finding Area Under a Curve | A-Level数学：曲线下面积的定积分求法

The definite integral is one of the most powerful tools in calculus, with a direct geometrical interpretation: it calculates the net signed area between a curve and the x-axis over a specified interval. For Edexcel A-Level Mathematics, mastering this technique is essential for Paper 1 (Pure Mathematics) and frequently appears in questions that combine algebraic manipulation, sketching curves, and evaluating integrals.

定积分是微积分中最强大的工具之一，它有着直接的几何意义：计算一条曲线与x轴之间在指定区间上的净符号面积。对于Edexcel A-Level数学，掌握这一技巧是Paper 1（纯数学）的关键，并且经常出现在结合代数运算、绘制曲线和计算积分的题目中。

1. The Fundamental Theorem of Calculus | 微积分基本定理

The link between differentiation and integration is expressed by the Fundamental Theorem of Calculus. If F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b equals F(b) − F(a).

微分与积分之间的联系由微积分基本定理表达。如果F(x)是f(x)的一个原函数，那么f(x)从a到b的定积分等于F(b) − F(a)。

∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)

This theorem allows us to evaluate area without approximating with rectangles. It is crucial to remember that the result is a number, not a function, and it can be positive, negative, or zero.

这一定理使我们无需通过矩形逼近就能计算面积。务必记住，结果是数值，而不是函数，且可以为正、为负或为零。

2. Standard Integrals You Must Know | 必须掌握的标准积分

To apply the Fundamental Theorem efficiently, you need quick recall of standard results. The Edexcel formula booklet provides these, but knowing them instinctively saves valuable exam time.

为了高效运用基本定理，你需要快速回忆标准结果。Edexcel公式册提供了这些公式，但本能地记住它们可以节省宝贵的考试时间。

Polynomials: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.
多项式：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中n ≠ −1。
Exponential: ∫ eˣ dx = eˣ + C.
指数函数：∫ eˣ dx = eˣ + C。
Reciprocal: ∫ (1/x) dx = ln|x| + C.
倒数函数：∫ (1/x) dx = ln|x| + C。
Trigonometric: ∫ cos x dx = sin x + C; ∫ sin x dx = −cos x + C.
三角函数：∫ cos x dx = sin x + C；∫ sin x dx = −cos x + C。

When evaluating between limits, the constant of integration C cancels out, so you may omit it during definite integration.

在代入上下限计算时，积分常数C会相互抵消，因此在定积分过程中可以省略C。

3. Area Under a Curve: The Positive Case | 曲线下的面积：正值情形

When a curve lies entirely above the x-axis for the interval [a, b], the area is simply the definite integral:

当曲线在区间[a, b]内完全位于x轴上方时，面积就是定积分：

Area = ∫ₐᵇ f(x) dx

Example: Find the area under y = x² from x = 1 to x = 3.

例题：求曲线y = x²在x = 1到x = 3之间与x轴围成的面积。

Area = ∫₁³ x² dx = [x³/3]₁³ = (27/3) − (1/3) = 26/3 ≈ 8.67

Since x² is always non-negative, this integral gives the true geometric area directly. Always check the sign of f(x) over the interval before assuming the integral equals the area.

由于x²始终非负，该积分直接给出真实的几何面积。在假设积分等于面积之前，务必先检查f(x)在区间上的符号。

4. Handling Negative Area: Curves Below the x-Axis | 处理负面积：曲线位于x轴下方

If a curve lies entirely below the x-axis over [a, b], the definite integral yields a negative value. Since area is a positive quantity, we take the absolute value:

如果曲线在[a, b]上完全位于x轴下方，定积分结果为负。由于面积是正值，我们取绝对值：

Area = |∫ₐᵇ f(x) dx|

Example: Find the area bounded by y = x² − 4 and the x-axis between x = 0 and x = 2.

例题：求曲线y = x² − 4与x轴在x = 0到x = 2之间围成的面积。

∫₀² (x² − 4) dx = [x³/3 − 4x]₀² = (8/3 − 8) − 0 = −16/3

Since the result is negative, the actual area is 16/3. Remember: the sign of the integral tells you which side of the x-axis the region lies.

由于结果为负，实际面积为16/3。记住：积分的符号告诉你区域位于x轴的哪一侧。

5. Regions Partly Above and Below the x-Axis | 横跨x轴上下的区域

When a curve crosses the x-axis within the integration interval, the integral alone would give the net signed area, which could be misleading. Areas above the x-axis and below must be computed separately and added.

当曲线在积分区间内穿过x轴时，仅凭积分会得到净符号面积，这可能产生误导。必须分别计算x轴上方和下方的面积，然后相加。

Example: Find the area between y = x(x − 2) and the x-axis from x = 0 to x = 3.

例题：求曲线y = x(x − 2)与x轴在x = 0到x = 3之间围成的面积。

First, find the roots of the function:

首先求函数的零点：

x(x − 2) = 0 → x = 0, x = 2

Thus, the curve is below the x-axis between 0 and 2, and above between 2 and 3. We split the integral:

因此，曲线在0到2之间位于x轴下方，在2到3之间位于x轴上方。我们拆分积分：

Area = |∫₀² (x² − 2x) dx| + ∫₂³ (x² − 2x) dx

= |(−4/3)| + (9 − 9) − (8/3 − 4) = 4/3 + 1/3 = 5/3

This approach ensures the total geometric area is correct. Neglecting to split the integral is a common exam pitfall.

这种方法确保总几何面积正确。忘记拆积分是考试中常见的陷阱。

6. Area Between a Curve and the y-Axis | 曲线与y轴之间的面积

Occasionally, questions require integrating with respect to y. If x can be expressed as a function of y, say x = g(y), then the area between the curve and the y-axis from y = c to y = d is:

偶尔，题目要求对y进行积分。如果x可以表示为y的函数，比如x = g(y)，那么曲线与y轴从y = c到y = d之间的面积为：

Area = ∫꜀ᵈ g(y) dy

Example: Find the area enclosed by y = x², the y-axis, and the horizontal line y = 4.

例题：求由y = x²、y轴和水平线y = 4围成的面积。

Rearrange to x = √y. Then:

改写为x = √y。于是：

Area = ∫₀⁴ √y dy = [2/3 y^(3/2)]₀⁴ = (2/3)(8) = 16/3

This technique is especially useful when the curve is more easily integrated with respect to y.

当曲线更容易对y积分时，这种技术尤其有用。

7. Area Between Two Curves | 两条曲线之间的面积

To find the area enclosed by two curves f(x) and g(x), first determine their intersection points. Then integrate the difference between the upper curve and the lower curve:

要求两条曲线f(x)和g(x)围成的面积，首先确定它们的交点，然后积分上曲线与下曲线之差：

Area = ∫ₐᵇ [f(x) − g(x)] dx

where f(x) ≥ g(x) on [a, b].

其中在[a, b]上f(x) ≥ g(x)。

Example: Find the area enclosed by y = x² and y = x + 2.

例题：求由y = x²和y = x + 2围成的面积。

Intersection points:

交点：

x² = x + 2 → x² − x − 2 = 0 → (x − 2)(x + 1) = 0 → x = −1, x = 2

The line y = x + 2 is above y = x² on [−1, 2]. Therefore:

直线y = x + 2在[−1, 2]上位于y = x²上方。因此：

Area = ∫₋₁² [(x + 2) − x²] dx = [x²/2 + 2x − x³/3]₋₁² = (2 + 4 − 8/3) − (1/2 − 2 + 1/3) = 9/2

Always sketch or mentally visualise the situation to determine which function is on top. If the curves cross inside the interval, split the region at the intersection points.

始终通过草图或想象判断哪个函数在上方。如果曲线在区间内相交，则在交点处拆分区域分别计算。

8. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Students frequently lose marks on area questions due to a few repeating errors. Being aware of these pitfalls before the exam can save you significant credit.

学生因为在面积题中反复犯几种错误而失分。考前意识到这些陷阱可以帮你挽回不少分数。

Forgetting to split the integral when the curve crosses the x-axis. This leads to a wrong “net” area instead of total area.
忘记拆分积分当曲线穿过x轴时，这会导致得到错误的”净”面积而非总面积。
Mistaking the upper and lower curves in a two-curve problem, which gives a negative or incorrect area.
搞混上曲线和下曲线在两曲线问题中，这会产生负值或错误的面积。
Algebraic errors when substituting limits, especially with negative numbers. Always use brackets when substituting.
代入上下限时的代数错误，特别是在负数情况下。代入时始终加括号。
Ignoring absolute value when the integrand is negative over the entire interval.
忽略绝对值当被积函数在整个区间上为负时。

Tip: For the Edexcel exam, even when questions don’t explicitly ask for a sketch, drawing a quick graph can help you plan the correct integration strategy.

提示：在Edexcel考试中，即使题目没有明确要求画图，快速画一个草图也能帮助你规划正确的积分策略。

9. Worked Examination-Style Question | 考试风格例题精讲

Let’s work through a complete Edexcel-style question that combines several skills we’ve discussed.

让我们一起完整地做一道结合了我们讨论的多个技能的Edexcel风格题目。

Question: The curve C has equation y = 3x − x². The line L has equation y = 2x. Find the area of the region enclosed by C, L, and the x-axis.

题目：曲线C的方程为y = 3x − x²。直线L的方程为y = 2x。求由C、L和x轴围成的区域面积。

Step 1: Understand the region. Sketch the curves and identify the enclosed area.

第一步：理解区域。画出曲线，识别围成的区域。

Curve C is a downward parabola. Line L passes through the origin with slope 2. Intersection points:

曲线C是开口向下的抛物线。直线L过原点且斜率为2。交点：

3x − x² = 2x → x − x² = 0 → x(1 − x) = 0 → x = 0, x = 1

The x-axis also bounds the region. The curve C intersects the x-axis at x = 0 and x = 3.

x轴也限定了该区域。曲线C在x = 0和x = 3处与x轴相交。

Step 2: Split the region. The enclosed area consists of two parts:

第二步：拆分区域。围成的区域由两部分组成：

Area = ∫₀¹ (2x − x) dx + ∫₁³ (3x − x²) dx

Step 3: Evaluate.

第三步：计算。

= ∫₀¹ x dx + [3x²/2 − x³/3]₁³

= [x²/2]₀¹ + ((27/2 − 9) − (3/2 − 1/3))

= 1/2 + (9/2) − (7/6) = 1/2 + 27/6 − 7/6 = 1/2 + 20/6 = 1/2 + 10/3 = 23/6

A clear diagram is essential for correctly identifying these boundary points and avoiding area overlap.

清晰的图形对于正确识别这些边界点、避免面积重叠至关重要。

10. Practice Problems | 练习题目

Test your understanding with these short problems.

用这些简短的题目来测试你的理解。

Problem 1: Find the area bounded by y = x² + 1, the x-axis, x = 1, and x = 3.
练习1：求由y = x² + 1、x轴、x = 1和x = 3围成的面积。
Problem 2: Calculate the area between y = sin x and the x-axis from x = 0 to x = 2π.
练习2：计算y = sin x与x轴从x = 0到x = 2π之间的面积。
Problem 3: Find the area enclosed by y = x³ and y = √x.
练习3：求由y = x³和y = √x围成的面积。

Answers: 1) 26/3. 2) 4 (split at x = π). 3) 5/12 (intersection at x = 0 and x = 1; upper curve is y = √x).

答案：1) 26/3。2) 4（在x = π处拆分）。3) 5/12（交点为x = 0和x = 1；上曲线是y = √x）。

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A-Level Mathematics: Definite Integrals for Finding Area Under a Curve | A-Level数学：曲线下面积的定积分求法