Parametric Equations & Integration | 参数方程与积分

📚 Parametric Equations & Integration | 参数方程与积分

Parametric equations are a cornerstone of A-Level Edexcel Pure Mathematics. In this revision article, we explore how to differentiate and integrate curves defined by a parameter, with exam-style techniques and common pitfalls clearly explained.

参数方程是A-Level爱德思纯数部分的核心内容。本篇复习文章将围绕以参数定义的曲线，深入讲解如何求导与积分，并给出考试风格的解题技巧与常见易错点。

1. What Are Parametric Equations? | 什么是参数方程？

A parametric curve is defined by expressing both x and y in terms of a third variable, usually t. Instead of writing y = f(x), we write x = f(t) and y = g(t). The variable t is called the parameter, and it often represents time or an angle.

参数曲线通过引入第三个变量（通常记为t）同时表示x和y。与直接写y = f(x)不同，我们写x = f(t)和y = g(t)。变量t称为参数，它常代表时间或角度。

For example, the curve defined by x = t², y = t³ describes a relationship between x and y that can be visualised by plotting points for various values of t.

例如，由x = t²，y = t³定义的曲线，可以通过为不同的t值描点来观察x与y之间的关系。

x = f(t), y = g(t)

2. Converting to Cartesian Form | 化为直角坐标形式

Sometimes it is useful to eliminate the parameter t to obtain a Cartesian equation in x and y. This is done by rearranging one equation to express t in terms of x or y, then substituting into the other equation.

有时我们需要消去参数t，得到只含x和y的直角坐标方程。方法是先从一个方程中解出t，再代入另一个方程。

Consider x = t² and y = t³. Since t = √x, substituting gives y = x^(3/2). Another common example is x = 2t, y = t² – 1: here t = x/2, so y = x²/4 – 1.

考虑x = t²和y = t³。由于t = √x，代入得y = x^(3/2)。另一个常见例子是x = 2t，y = t² – 1：这里t = x/2，因此y = x²/4 – 1。

When trigonometric functions are involved, the identity sin²θ + cos²θ = 1 is often the key to eliminating t.

当涉及三角函数时，恒等式sin²θ + cos²θ = 1通常是消去参数t的关键。

3. Differentiation of Parametric Curves | 参数曲线求导

To find dy/dx for a parametric curve, we use the chain rule:

对于参数曲线求dy/dx，我们使用链式法则：

dy/dx = (dy/dt) ÷ (dx/dt)

This formula is valid provided dx/dt ≠ 0. It allows us to find the gradient at any point without eliminating t.

此公式在dx/dt ≠ 0时成立。它允许我们无需消去t即可求出任意一点的梯度。

Example: For x = t², y = t³, we have dx/dt = 2t and dy/dt = 3t². Therefore dy/dx = 3t² / 2t = 3t/2.

示例：对于x = t²，y = t³，有dx/dt = 2t，dy/dt = 3t²。因此dy/dx = 3t² / 2t = 3t/2。

At t = 1, the gradient is 3/2; at t = 0, dx/dt = 0, so the gradient is undefined and the curve may have a vertical tangent.

在t = 1处，梯度为3/2；在t = 0处，dx/dt = 0，因此梯度未定义，曲线可能存在垂直切线。

4. Second Derivatives | 二阶导数

To find d²y/dx², we differentiate dy/dx with respect to t and divide by dx/dt again:

要求d²y/dx²，我们需要对dy/dx关于t求导，再除以dx/dt：

d²y/dx² = d/dt(dy/dx) ÷ dx/dt

For x = t², y = t³, we had dy/dx = 3t/2. Then d/dt(3t/2) = 3/2. Dividing by dx/dt = 2t gives d²y/dx² = 3/(4t).

对于x = t²，y = t³，已有dy/dx = 3t/2。则d/dt(3t/2) = 3/2。除以dx/dt = 2t得d²y/dx² = 3/(4t)。

This allows us to determine the concavity of the curve or find stationary points. Stationary points occur where dy/dx = 0, i.e. where dy/dt = 0 but dx/dt ≠ 0.

这使我们能够判断曲线的凹凸性或求驻点。驻点出现在dy/dx = 0时，即dy/dt = 0且dx/dt ≠ 0。

5. Integration: Area under a Parametric Curve | 积分：参数曲线下的面积

The area between a parametric curve and the x-axis can be found using the substitution x = f(t):

参数曲线与x轴之间的面积可以通过代换x = f(t)求得：

Area = ∫ y dx = ∫ y (dx/dt) dt

The limits of integration must be converted from x-values to corresponding t-values.

积分的上下限必须从x值转换为对应的t值。

Example: Find the area under x = t², y = t³ from t = 0 to t = 1. Since x = t², the x-limits are 0 to 1. Thus Area = ∫₀¹ t³ · (2t) dt = ∫₀¹ 2t⁴ dt = 2/5.

示例：求x = t²，y = t³从t = 0到t = 1曲线下的面积。由于x = t²，x的上下限为0到1。因此面积 = ∫₀¹ t³ · (2t) dt = ∫₀¹ 2t⁴ dt = 2/5。

Note: If the curve goes below the x-axis, the integral may be negative. Always sketch the curve to check the sign.

注意：如果曲线位于x轴下方，积分结果可能为负。应始终画出曲线草图以检查符号。

6. Area Enclosed by a Parametric Curve | 参数曲线围成的面积

When a curve forms a closed loop, the enclosed area can be computed directly using:

当曲线形成闭合环路时，围成的面积可以直接用以下公式计算：

Area = ∫ y dx = ∫ y (dx/dt) dt = -∫ x (dy/dt) dt

For a closed curve traversed anticlockwise, both forms give the same positive result. The choice depends on which integral is easier to evaluate.

对于逆时针方向行进的闭合曲线，两种形式给出相同的正值。选择哪一种取决于哪个积分更容易计算。

Example: Find the area enclosed by x = cos t, y = sin t for 0 ≤ t ≤ 2π. We have dx/dt = -sin t, so Area = ∫₀^(2π) sin t · (-sin t) dt = -∫₀^(2π) sin²t dt = -π. Taking the absolute value gives π, which is the area of a unit circle.

示例：求x = cos t，y = sin t在0 ≤ t ≤ 2π范围内围成的面积。有dx/dt = -sin t，所以面积 = ∫₀^(2π) sin t · (-sin t) dt = -∫₀^(2π) sin²t dt = -π。取绝对值得到π，即单位圆的面积。

7. Integrating y·(dx/dt) with Limits | 带上下限的y·(dx/dt)积分

A common exam question provides x as a function of t, y as a function of t, and asks for the area under the curve between two x-values. The key steps are:

一种常见的考试题型是给出x(t)和y(t)，要求计算两个x值之间曲线下的面积。关键步骤如下：

Find dx/dt from x(t).
Express the integrand as y(t) · dx/dt.
Convert the x-limits to t-limits using x = f(t).
Evaluate the integral with respect to t.

由x(t)求出dx/dt。
将被积函数表示为y(t) · dx/dt。
用x = f(t)将x的上下限转换为t的上下限。
对t进行积分。

For example, if x = 2t, y = t² and the area is required between x = 0 and x = 4, then t = x/2, so t = 0 and t = 2 are the new limits. Area = ∫₀² t² · 2 dt = 2·(t³/3)|₀² = 16/3.

例如，若x = 2t，y = t²，需要求x = 0到x = 4之间的面积，则t = x/2，新上下限为t = 0和t = 2。面积 = ∫₀² t² · 2 dt = 2·(t³/3)|₀² = 16/3。

8. Arc Length of a Parametric Curve | 参数曲线的弧长

The length of a curve between t = t₁ and t = t₂ is given by:

曲线从t = t₁到t = t₂的长度由下式给出：

Arc length = ∫ √((dx/dt)² + (dy/dt)²) dt

This formula comes from Pythagoras’ theorem applied to infinitesimal increments. It is used when the question explicitly asks for arc length or when deriving the surface area of revolution.

该公式源于对无穷小增量应用勾股定理。当题目明确要求弧长或推导旋转体表面积时会用到。

Example: For x = cos t, y = sin t, 0 ≤ t ≤ π, we have dx/dt = -sin t, dy/dt = cos t. The arc length is ∫₀^π √(sin²t + cos²t) dt = ∫₀^π 1 dt = π. This is half the circumference of a unit circle.

示例：对于x = cos t，y = sin t，0 ≤ t ≤ π，有dx/dt = -sin t，dy/dt = cos t。弧长为∫₀^π √(sin²t + cos²t) dt = ∫₀^π 1 dt = π。这是单位圆周长的一半。

9. Surface Area of Revolution | 旋转体表面积

When a parametric curve is rotated about the x-axis, the surface area generated is:

当参数曲线绕x轴旋转时，生成的曲面面积为：

Surface area = 2π ∫ y √((dx/dt)² + (dy/dt)²) dt

If the rotation is about the y-axis, replace y by x:

如果绕y轴旋转，则将y替换为x：

Surface area = 2π ∫ x √((dx/dt)² + (dy/dt)²) dt

In both cases, the limits must be chosen according to the t-values that correspond to the rotating arc.

两种情况下，上下限都需根据旋转弧对应的t值选取。

10. Exam-Style Worked Example | 考试风格例题

Question: A curve is defined by x = t² – 1, y = t³ – 3t for t ≥ 0. Find the area of the finite region enclosed by the curve and the x-axis.

题目：曲线由x = t² – 1，y = t³ – 3t（t ≥ 0）定义。求该曲线与x轴围成的有限区域的面积。

Solution: The curve meets the x-axis when y = 0, i.e. t³ – 3t = 0, so t(t² – 3) = 0. Since t ≥ 0, we have t = 0 or t = √3.

解答：曲线与x轴相交于y = 0时，即t³ – 3t = 0，因此t(t² – 3) = 0。由于t ≥ 0，得t = 0或t = √3。

We compute dx/dt = 2t. The area is:

计算dx/dt = 2t。面积为：

Area = ∫₀^(√3) (t³ – 3t)(2t) dt = ∫₀^(√3) (2t⁴ – 6t²) dt

Evaluating: [2t⁵/5 – 2t³]₀^(√3) = (2·(√3)⁵/5 – 2·(√3)³) – 0 = 2·3²·√3/5 – 2·3√3 = (18√3/5) – 6√3 = (18√3 – 30√3)/5 = -12√3/5.

求值：[2t⁵/5 – 2t³]₀^(√3) = (2·(√3)⁵/5 – 2·(√3)³) – 0 = 2·3²·√3/5 – 2·3√3 = (18√3/5) – 6√3 = (18√3 – 30√3)/5 = -12√3/5。

Since the region lies below the x-axis, the area is the absolute value: 12√3/5 square units.

由于该区域位于x轴下方，面积为绝对值：12√3/5 平方单位。

11. Common Pitfalls and Tips | 常见易错点与技巧

Several common mistakes appear regularly in exams. Understanding these can save valuable marks.

考试中经常出现一些常见错误。理解这些错误可以帮你保住宝贵的分数。

Forgetting to convert the limits when changing from x to t. Always verify the correspondence.
Using dy/dx = dy/dt ÷ dx/dt but omitting the condition dx/dt ≠ 0. Check this before dividing.
Multiplying by dx/dt but leaving t in terms of x. The integrand must be entirely in terms of t before integrating.
Ignoring the sign of the area. Sketch the curve or check the sign of y(t) over the range.
Forgetting the factor 2π in surface area formulas.

从x换成t时忘记转换上下限。始终验证对应关系。
使用dy/dx = dy/dt ÷ dx/dt时忽略dx/dt ≠ 0的条件。除法前先检查。
乘以dx/dt后仍保留x表达t的形式。积分前被积函数必须完全用t表示。
忽略面积的符号。画出草图或检查y(t)在区间内的符号。
在表面积公式中漏掉因子2π。

Formula	When to Use
dy/dx = (dy/dt)/(dx/dt)	Gradient of a parametric curve
Area = ∫ y(dx/dt) dt	Area under a curve or between curve and axis
Arc length = ∫ √((dx/dt)² + (dy/dt)²) dt	Length of a curve segment
Surface area = 2π ∫ y·√((dx/dt)² + (dy/dt)²) dt	Rotation about x-axis

12. Practice Questions | 练习题

Try these problems on your own before checking the answers.

先独立尝试以下练习，再核对答案。

1. A curve is defined by x = 2t + 1, y = t² – t. Find dy/dx at t = 1.

1. 曲线由x = 2t + 1，y = t² – t定义。求t = 1处的dy/dx。

2. Find the area enclosed by x = 2cos t, y = 3sin t for 0 ≤ t ≤ 2π.

2. 求x = 2cos t，y = 3sin t在0 ≤ t ≤ 2π范围内围成的面积。

3. Find the arc length of x = t², y = 2t from t = 0 to t = 2.

3. 求x = t²，y = 2t从t = 0到t = 2的弧长。

Answers: 1. dy/dx = (2t – 1)/2, at t = 1 gives 1/2. 2. Area = ∫₀^(2π) 3sin t · (-2sin t) dt = -6π, so area = 6π. 3. Arc length = ∫₀² √((2t)² + 4) dt = ∫₀² 2√(t² + 1) dt ≈ 2[(t/2)√(t²+1) + (1/2)sinh⁻¹(t)]₀² ≈ 6.14.

答案：1. dy/dx = (2t – 1)/2，在t = 1时为1/2。2. 面积 = ∫₀^(2π) 3sin t · (-2sin t) dt = -6π，因此面积为6π。3. 弧长 = ∫₀² √((2t)² + 4) dt = ∫₀² 2√(t² + 1) dt ≈ 2[(t/2)√(t²+1) + (1/2)sinh⁻¹(t)]₀² ≈ 6.14。

Mastering parametric integration is about practice. Always write down dx/dt explicitly, check the limits, and verify whether the area is positive or negative. With these steps, you are ready for any Edexcel exam question on this topic.

掌握参数积分的关键在于练习。始终明确写出dx/dt，检查上下限，并验证面积为正还是负。有了这些步骤，你就能应对爱德思考试中关于此考点的任何题目。

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Parametric Equations & Integration | 参数方程与积分