The Area Bounded by a Polar Curve | 极坐标曲线围成的面积

📚 The Area Bounded by a Polar Curve | 极坐标曲线围成的面积

In AQA A-Level Mathematics, the polar coordinate topic includes finding the area enclosed by a polar curve. This article explains where the formula comes from, how to set up limits correctly, and how to avoid common exam errors.

在 AQA A-Level 数学中，极坐标部分包括求由极坐标曲线围成的面积。本文解释面积公式的来源、如何正确设定积分限，以及如何避免考试中的常见错误。

1. Polar Coordinates Recap | 极坐标回顾

A point P can be described by polar coordinates (r, θ), where r is the distance from the origin and θ is the angle measured anticlockwise from the initial line.

一个点 P 可以用极坐标 (r, θ) 表示，其中 r 是到原点的距离，θ 是从极轴逆时针测量的角度。

In Cartesian form, x = r cos θ and y = r sin θ, so r² = x² + y².

在直角坐标形式下，x = r cos θ，y = r sin θ，因此 r² = x² + y²。

Many AQA questions give r as a function of θ, for example r = a(1 + cos θ).

许多 AQA 试题给出 r 关于 θ 的函数，例如 r = a(1 + cos θ)。

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2. The Sector Area Formula | 扇形面积公式

The region enclosed by a polar curve is built from thin sectors instead of vertical strips. A sector of angle dθ and radius r has area approximately ½ r² dθ.

极坐标曲线围成的区域由许多细小的扇形组成，而不是竖直条带。一个角度为 dθ、半径为 r 的扇形面积近似为 ½ r² dθ。

Integrating these sectors from θ = α to θ = β gives the key formula:

将这些扇形从 θ = α 到 θ = β 积分，得到关键公式：

A = ½ ∫ r² dθ

Here r is the polar function, and the integral is evaluated from the starting angle α to the ending angle β.

这里 r 是极坐标函数，积分从起始角 α 到终止角 β 计算。

The formula is in radians only, because the sector area ½ r² θ is only valid when θ is measured in radians.

该公式仅适用于弧度制，因为扇形面积 ½ r² θ 只有在 θ 用弧度表示时才成立。

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3. Setting Up the Integral | 建立积分式

Before integrating, you must identify the polar curve and the region whose area is required. In simple cases, the curve starts and ends at the origin, or forms a closed loop.

在积分之前，你必须先确定极坐标曲线以及所求面积的区域。在简单情形中，曲线从原点开始并回到原点，或者形成一个闭合环。

Write r = f(θ), square it, and substitute into A = ½ ∫ r² dθ. Then choose the limits α and β so that the region is swept exactly once.

写出 r = f(θ)，将其平方后代入 A = ½ ∫ r² dθ。然后选择限 α 和 β，使该区域恰好被扫过一次。

For example, if r = 2θ for 0 ≤ θ ≤ π, the area of the swept region is A = ½ ∫₀^π (2θ)² dθ = ½ ∫₀^π 4θ² dθ.

例如，若 r = 2θ，0 ≤ θ ≤ π，则扫过区域的面积为 A = ½ ∫₀^π (2θ)² dθ = ½ ∫₀^π 4θ² dθ。

Always simplify r² before integrating, and state the formula clearly in the exam.

在考试中一定要先化简 r² 再积分，并清楚地写出公式。

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4. Choosing Limits of Integration | 选择积分限

Finding the correct θ-limits is often the hardest part. For a closed curve, solve r = 0 to find where the curve passes through the pole.

找到正确的 θ 限通常是最难的部分。对于闭合曲线，解 r = 0 可以找到曲线经过极点的位置。

For r = a(1 + cos θ), solving a(1 + cos θ) = 0 gives cos θ = -1, so θ = π. The full curve is traced from 0 to 2π, but the area can often be doubled using symmetry.

对于 r = a(1 + cos θ)，解 a(1 + cos θ) = 0 得 cos θ = -1，因此 θ = π。完整曲线从 0 到 2π 描出，但面积通常可利用对称性翻倍计算。

Check that the limits trace the intended region exactly once. If the curve retraces itself, integrating over the full range may double-count or cancel incorrectly.

检查积分限是否恰好描出目标区域一次。如果曲线重复经过自身，对整个范围积分可能会重复计算或错误抵消。

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5. Handling Symmetry | 处理对称性

Many polar curves used in AQA exams have symmetry about the initial line or other axes. Using symmetry can reduce the integration work.

AQA 考试中许多极坐标曲线关于极轴或其他轴对称。利用对称性可以减少积分计算量。

For example, r = a cos θ is a circle symmetric about the initial line. Its full area can be found by integrating from 0 to π/2 and then multiplying by 2.

例如，r = a cos θ 是一个关于极轴对称的圆。可以先从 0 到 π/2 积分，然后乘以 2 得到完整面积。

If r = a sin θ, the circle is symmetric about the vertical line θ = π/2. You may integrate from 0 to π/2 and multiply by 2.

若 r = a sin θ，圆关于竖直线 θ = π/2 对称。你可以从 0 到 π/2 积分并乘以 2。

Always justify any doubling by stating which symmetry you are using.

任何翻倍操作都要说明你所使用的对称性。

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6. Loops and Petals | 环与花瓣

Curves such as r = a cos nθ or r = a sin nθ produce loops or petals. The number of petals depends on n.

r = a cos nθ 或 r = a sin nθ 等曲线会产生环或花瓣。花瓣的数量取决于 n。

When n is odd, the curve has n petals. When n is even, it has 2n petals. Each petal is swept as r goes from 0 to 0 again.

当 n 为奇数时，曲线有 n 个花瓣；当 n 为偶数时，有 2n 个花瓣。每个花瓣在 r 从 0 到 0 的过程中扫出。

For r = a cos 3θ, one petal is traced for θ from -π/6 to π/6, and the total area is 3 times the area of one petal.

对于 r = a cos 3θ，一个花瓣在 θ 从 -π/6 到 π/6 时描出，总面积为单个花瓣面积的 3 倍。

To find limits for one petal, solve r = 0 and choose two consecutive solutions that enclose the petal.

要找到一个花瓣的限，解 r = 0 并选择两个相邻的解，它们包围该花瓣。

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7. Area Between Two Polar Curves | 两曲线之间的面积

If a region is bounded by two polar curves r = r₁(θ) and r = r₂(θ), the area between them is given by:

如果区域由两条极坐标曲线 r = r₁(θ) 和 r = r₂(θ) 围成，则它们之间的面积为：

A = ½ ∫

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