📚 Parametric Equations: Differentiation & Integration | 参数方程：微分与积分
Parametric equations allow us to describe curves in the plane by introducing an independent parameter, often denoted by t. In Edexcel A-Level Mathematics, candidates are required to differentiate and integrate functions defined parametrically, find tangents and normals, compute areas and volumes of revolution, and interpret parametric forms in kinematics. This revision guide covers every essential technique, highlights common errors, and provides a structured approach to exam questions.
参数方程通过引入一个独立参数（通常记为 t）来描述平面上的曲线。在 Edexcel A-Level 数学中，考生需要掌握参数函数的微分与积分、求切线和法线、计算面积和旋转体体积，并在运动学中理解参数形式。本复习指南涵盖每个关键技巧，指出常见错误，并为应试提供有条理的方法。
1. Introduction to Parametric Equations | 参数方程简介
pecially useful for curves that are not functions in the usual sense, such as loops and self-intersections. Classic examples include the circle x = r cos t, y = r sin t and the parabola x = t, y = t².
参数曲线由 x = f(t)、y = g(t) 给出，t 属于某个定义域。随着参数 t 变化，点 (x, y) 描绘出一条轨迹。这种表示法对于通常意义下不是函数的曲线（如回环和自交曲线）特别有用。经典例子包括圆 x = r cos t、y = r sin t 和抛物线 x = t、y = t²。
At A-Level, you will often be given parametric equations and asked to work directly with them without eliminating the parameter. Understanding the relationship between t, x and y is the foundation for both differentiation and integration.
在 A-Level 中，通常会给出参数方程，并要求直接使用它们而不消去参数。理解 t、x 和 y 之间的关系是微分与积分的基础。
2. Eliminating the Parameter | 消去参数
ons are involved. For instance, from x = 2 cos t, y = 2 sin t we can use cos² t + sin² t = 1 to obtain x² + y² = 4. For algebraic forms, isolate t from one equation and substitute into the other.
要从参数形式转化为直角坐标形式，需在两条方程之间消去 t。如果涉及三角函数，可使用代入法或三角恒等式。例如，由 x = 2 cos t、y = 2 sin t，利用 cos² t + sin² t = 1 可得 x² + y² = 4。对于代数形式，可从一个方程中解出 t 并代入另一方程。
While eliminating the parameter is not always required for calculus, it can help with sketching and understanding the overall shape of the curve. Be aware that the domain of t may restrict the Cartesian curve to only a portion of the familiar shape.
尽管微积分计算并不总需要消去参数，但这样做有助于绘制草图和理解曲线的整体形状。注意，t 的定义域可能会将直角坐标曲线限制为熟悉形状的一部分。
3. Sketching Parametric Curves | 绘制参数曲线
tion and any turning points. Indicate the orientation with arrows pointing in the direction of increasing t. For trigonometric parameters, picking t = 0, π/2, π, 3π/2 often captures the main features.
要绘制参数曲线，可在给定的 t 区间上构造 t、x 和 y 的取值表。选取能揭示运动方向和任何转折点的关键 t 值。用箭头标出 t 增大的方向。对于三角函数参数，选取 t = 0、π/2、π、3π/2 通常能捕捉主要特征。
When the curve has a restricted domain, mark the start and end points clearly. If the curve intersects itself, identify the t-values at which this occurs. A good sketch is invaluable for setting up area or volume integrals correctly.
当曲线具有受限定义域时，清晰标出起点和终点。如果曲线自交，找出发生自交的 t 值。一幅良好的草图对于正确设置面积或体积积分至关重要。
4. Chain Rule – First Derivative | 链式法则与一阶导数
The gradient of a parametric curve is found by differentiating y with respect to x, which requires the chain rule. The fundamental formula is
参数曲线的斜率通过对 x 求导 y 得出，这需要使用链式法则。基本公式为
dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.
You must compute dx/dt and dy/dt separately, then form their quotient. Always simplify the result and express it in terms of t unless the question asks for a numerical value at a particular point.
你必须分别计算 dx/dt 和 dy/dt，然后构造商。除非题目要求特定点的数值，否则应化简结果并将其表示为 t 的函数。
= t³ + t, y = 2t² − 1, then dx/dt = 3t² + 1 and dy/dt = 4t. Hence dy/dx = 4t / (3t² + 1).
例如，若 x = t³ + t，y = 2t² − 1，则 dx/dt = 3t² + 1，dy/dt = 4t。因此 dy/dx = 4t / (3t² + 1)。
5. Tangents and Normals | 切线与法线
tes (x(a), y(a)) and use y − y₁ = m(x − x₁). The normal line has gradient −1/m (if m ≠ 0).
要求 t = a 对应点处的切线方程，首先计算 t = a 时的 dy/dx 得到斜率 m。然后求出坐标 (x(a), y(a))，并利用 y − y₁ = m(x − x₁)。法线的斜率为 −1/m（若 m ≠ 0）。
ded dy/dt ≠ 0). Identifying these helps with curve sketching and solving stationary point problems.
当 dy/dt = 0 且 dx/dt ≠ 0 时，出现水平切线；当 dx/dt = 0 且 dy/dt ≠ 0 时，出现竖直切线。识别这些有助于绘制曲线和求解驻点问题。
Examiners often award marks specifically for stating the chain rule formula and for correctly substituting the parameter. Always show the steps clearly.
考官通常会对写出链式法则公式和正确代入参数赋予步骤分。务必清晰地展示步骤。
6. Second Derivatives | 二阶导数
The second derivative d²y/dx² measures the rate of change of the gradient. In parametric form, it is obtained by differentiating dy/dx with respect to t and then dividing by dx/dt:
二阶导数 d²y/dx² 衡量斜率的变化率。在参数形式下，它通过先对 t 求导 dy/dx，再除以 dx/dt 得到：
d²y/dx² = (d/dt)[dy/dx] / (dx/dt).
This is a common source of mistakes: many students forget the division by dx/dt and simply differentiate dy/dx with respect to t, which gives d²y/dxdt, not the required second derivative. Always apply the full formula and keep the answer in terms of t.
这是一个常见的错误源：许多学生忘记除以 dx/dt，仅仅对 t 求导 dy/dx，得到的是 d²y/dxdt，而非所需的二阶导数。始终应用完整公式，并将答案保留为 t 的函数。
If the question asks for the nature of a stationary point, compute d²y/dx² at the relevant t-value. A negative value indicates a local maximum, a positive value a local minimum.
若题目要求判断驻点性质，计算相关 t 值处的 d²y/dx²。负值表明局部极大值，正值表明局部极小值。
7. Area Under a Parametric Curve | 参数曲线下的面积
y ∫ y dx. Substituting dx = (dx/dt) dt transforms the integral into one with respect to t:
参数曲线与 x 轴之间在 x 区间 [x₁, x₂] 上的面积由 ∫ y dx 给出。代入 dx = (dx/dt) dt 将积分转化为关于 t 的积分：
Area = ∫t₁t₂ y(t) · (dx/dt) dt.
nd to x₁ and x₂ respectively, i.e., x(t₁) = x₁ and x(t₂) = x₂. Be careful with orientation – he integral might produce a negative value; take the absolute value for the physical area.
积分限 t₁ 和 t₂ 必须分别对应 x₁ 和 x₂，即 x(t₁) = x₁、x(t₂) = x₂。注意方向——若 t 增大时 x 减小，积分可能会给出负值；应取绝对值作为实际面积。
ntegrate between each pair. The area of a loop is often calculated using symmetry to simplify the work.
对于回环，找出曲线与 x 轴相交（y = 0）的 t 值，并在每对 t 值之间积分。回环的面积常利用对称性来简化计算。
8. Volumes of Revolution | 旋转体体积
When a parametric curve is rotated about the x-axis, the volume of revolution is V = π ∫ab y² dx. In parametric form, this becomes
当参数曲线绕 x 轴旋转时，旋转体体积为 V = π ∫ab y² dx。参数形式下，这变为
V = π ∫t₁t₂ y² · (dx/dt) dt.
g: if y is negative, y² is positive, so no sign issues arise, but you must still expand and integrate correctly.
积分限 t₁ 和 t₂ 对应 x = a 和 x = b。平方时要仔细：若 y 为负，y² 为正，因此无符号问题，但仍需正确展开并积分。
case, use V = π ∫ x² dy with dy = (dy/dt) dt and limits in t corresponding to the y-range.
题目也可能要求绕 y 轴旋转，但这较少见。此时使用 V = π ∫ x² dy，其中 dy = (dy/dt) dt，积分限为 t 值对应 y 范围。
Always check whether the question intends the region to be bounded by the curve and coordinate axes, as a sketched diagram is rarely provided in the exam.
务必检查题目所围区域是否由曲线和坐标轴界定，因为考试中很少提供插图。
9. Parametric Equations in Kinematics | 参数方程在运动学中的应用
, dy/dt), and the speed is the magnitude √[(dx/dt)² + (dy/dt)²]. The acceleration vector components are the second derivatives d²x/dt² and d²y/dt².
在力学中，粒子在时刻 t 的位置通常由参数方程 x = f(t), y = g(t) 给出。速度向量分量为 (dx/dt, dy/dt)，速率的大小为 √[(dx/dt)² + (dy/dt)²]。加速度向量分量为二阶导数 d²x/dt² 和 d²y/dt²。
ions may require you to find the time when the particle is moving parallel to the x-axis (dy/dt = 0) or y-axis (dx/dt =
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