The Area Bounded by a Polar Curve | 极坐标曲线所围面积

📚 The Area Bounded by a Polar Curve | 极坐标曲线所围面积

In AQA A-Level Mathematics, the study of polar coordinates extends your understanding of integration to curves defined by \( r = f(\theta) \) — although we will write all formulae without LaTeX, using standard notation. The key skill is to calculate the area enclosed by a polar curve or between two polar curves, using the formula based on sectors.

在 AQA A-Level 数学中，极坐标的学习将积分技巧拓展到由 \( r = f(\theta) \) 定义的曲线。核心技能是使用基于扇形的公式，计算极坐标曲线所围成的面积或两条极坐标曲线之间的面积。下文的所有公式均使用标准 Unicode 记号，不涉及 LaTeX。

1. Polar Coordinates Refresher | 极坐标基础回顾

A point in the plane can be described by polar coordinates \( (r, \theta) \), where \( r \) is the distance from the pole (origin) and \( \theta \) is the angle measured anticlockwise from the initial line (positive x-axis). The conversion to Cartesian coordinates is \( x = r \cos \theta \), \( y = r \sin \theta \), so that \( r^2 = x^2 + y^2 \).

平面上的点可以用极坐标 \( (r, \theta) \) 表示，其中 \( r \) 是到极点（原点）的距离，\( \theta \) 是从极轴（正 x 轴）逆时针旋转的角度。转换为直角坐标为 \( x = r \cos \theta \)，\( y = r \sin \theta \)，因此 \( r^2 = x^2 + y^2 \)。

For a curve given by \( r = f(\theta) \), as \( \theta \) increases, the radius \( r \) varies, tracing a curve. Positive \( r \) means the point lies in the direction of \( \theta \); negative \( r \) means the point lies in the opposite direction (angle \( \theta + \pi \)).

对于由 \( r = f(\theta) \) 给出的曲线，当 \( \theta \) 增大时，半径 \( r \) 随之变化，从而描绘出曲线。正 \( r \) 表示点位于 \( \theta \) 方向上；负 \( r \) 表示点位于相反方向（角度 \( \theta + \pi \)）。

2. The Sector Area Formula | 扇形面积公式

The area of a sector with radius \( r \) and angle \( \mathrm{d}\theta \) is \( \frac{1}{2} r^2 \, \mathrm{d}\theta \). Summing such infinitesimal sectors between \( \theta = \alpha \) and \( \theta = \beta \) gives the exact area bounded by the polar curve:

半径为 \( r \)、角度为 \( \mathrm{d}\theta \) 的扇形面积为 \( \frac{1}{2} r^2 \, \mathrm{d}\theta \)。将这些无穷小扇形从 \( \theta = \alpha \) 到 \( \theta = \beta \) 求和，即可得到极坐标曲线所围的精确面积：

Area = ∫_α^β ½ r² dθ = ½ ∫_α^β [f(θ)]² dθ

This formula is the polar analogue of the Cartesian area under a curve. Notice the factor ½, which arises because each element is a triangle-like sector, not a rectangle.

该公式是直角坐标下曲线下面积的极坐标类比。注意系数 ½，这是因为每个面积微元是类似三角形的扇形，而不是矩形。

3. Conditions for Using the Formula | 公式的适用条件

To apply the formula correctly, the curve must be traced exactly once as \( \theta \) runs from \( \alpha \) to \( \beta \). The angle limits must correspond to the sector that encloses the required region.

要正确应用该公式，当 \( \theta \) 从 \( \alpha \) 到 \( \beta \) 变化时，曲线必须恰好被描绘一次。角度限必须对应包围所求区域的扇形部分。

If the curve is closed and passes through the pole, find the values of \( \theta \) where \( r = 0 \). These often become the integration limits.

如果曲线是封闭的且经过极点，则求 \( r = 0 \) 时对应的 \( \theta \) 值。这些值通常用作积分限。
If the region is bounded by two curves \( r = f(\theta) \) and \( r = g(\theta) \), the area between them is \( \frac{1}{2}\int [f^2 – g^2] \, d\theta \), taking the outer radius minus the inner radius.

如果区域由两条曲线 \( r = f(\theta) \) 和 \( r = g(\theta) \) 围成，则它们之间的面积为 \( \frac{1}{2}\int [f^2 – g^2] \, d\theta \)，取外半径平方减去内半径平方。
Always ensure the angle is measured in radians when integrating.

积分时角度必须使用弧度制。

4. Exploiting Symmetry | 利用对称性

Many polar curves exhibit symmetry about the initial line, the line \( \theta = \pi/2 \), or the pole. You can calculate the area of one symmetric part and then multiply by the number of identical parts.

许多极坐标曲线关于极轴、直线 \( \theta = \pi/2 \) 或极点具有对称性。你可以计算一个对称部分的面积，然后乘以相同部分的数量。

For example, the circle \( r = 2a \cos \theta \) is symmetric about the initial line. The full area is \( 2 \times \frac{1}{2}\int_{0}^{\pi/2} (2a \cos \theta)^2 \, d\theta \). This avoids integrating over a full range where \( r \) would be negative if you used 0 to 2π.

例如，圆 \( r = 2a \cos \theta \) 关于极轴对称。整个面积为 \( 2 \times \frac{1}{2}\int_{0}^{\pi/2} (2a \cos \theta)^2 \, d\theta \)。这样可以避免从 0 到 2π 积分时出现负 \( r \) 的问题。

AQA exam questions often expect you to identify a symmetrical lobe and multiply by 2 or 4, rather than integrating over a long interval.

AQA 考题常希望你识别对称的一叶，然后乘以 2 或 4，而不是在长区间上积分。

5. Finding Intersection Points | 求交点确定积分限

To find where two polar curves intersect, solve \( f(\theta) = g(\theta) \). However, be careful: the pole may also be an intersection even if no angle solves that equation, because curves may pass through the pole at different \( \theta \) values.

要求两条极坐标曲线的交点，解方程 \( f(\theta) = g(\theta) \)。但要注意：极点也可能是一个交点，即使没有某个角度能使该方程成立，因为两条曲线可能在不同的 \( \theta \) 值经过极点。

For a single closed curve, the values of \( \theta \) at which \( r = 0 \) are the tangents from the pole that bound each loop. For example, the rose curve \( r = \sin 2\theta \) has loops between \( \theta = 0 \) and \( \theta = \pi/2 \), then \( \theta = \pi \) and \( 3\pi/2 \).

对于单条封闭曲线，使 \( r = 0 \) 的 \( \theta \) 值是从极点引出的切线，它们界定了每一叶。例如，玫瑰线 \( r = \sin 2\theta \) 的一叶介于 \( \theta = 0 \) 和 \( \theta = \pi/2 \) 之间，另一叶介于 \( \theta = \pi \) 和 \( 3\pi/2 \) 之间。

6. Worked Example: Circle | 示例：圆

Find the area enclosed by the circle \( r = 4 \cos \theta \).

求圆 \( r = 4 \cos \theta \) 所围的面积。

The curve is a circle with diameter 4. It passes through the pole at \( \theta = \pi/2 \), and the maximum radius occurs at \( \theta = 0 \). Using symmetry about the initial line, the full area is twice the area from 0 to π/2:

该曲线是直径为 4 的圆。它在 \( \theta = \pi/2 \) 处经过极点，最大半径出现在 \( \theta = 0 \)。利用关于极轴的对称性，整个面积是 0 到 π/2 面积的两倍：

Area = 2 × ½ ∫₀^π/2 (4 cos θ)² dθ = ∫₀^π/2 16 cos² θ dθ

Using \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \), we get:

利用 \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \)，得：

Area = 16 × [ θ/2 + sin 2θ /4 ]₀^π/2 = 16 × π/4 = 4π

The actual circle has radius 2, so its area is \( \pi \times 2^2 = 4\pi \). Correct!

该圆实际半径为 2，所以面积是 \( \pi \times 2^2 = 4\pi \)。结果正确！

7. Cardioid Area | 心形线面积

The cardioid \( r = a(1 + \cos \theta) \) is a classic AQA example. To find its total area, note that the whole curve is traced for \( 0 \le \theta \le 2\pi \), and it is symmetric about the initial line.

心形线 \( r = a(1 + \cos \theta) \) 是 AQA 的经典例题。求其总面积时，注意整条曲线在 \( 0 \le \theta \le 2\pi \) 内被完整描绘，且关于极轴对称。

Using the full range is safe here because \( r \) is never negative; it ranges from 0 at \( \theta = \pi \) to \( 2a \) at \( \theta = 0 \).

这里使用完整区间是安全的，因为 \( r \) 从未为负；它的范围是从 \( \theta = \pi \) 时的 0 到 \( \theta = 0 \) 时的 \( 2a \)。

Area = ½ ∫₀^2π a² (1 + cos θ)² dθ = ½ a² ∫₀^2π (1 + 2 cos θ + cos² θ) dθ

With \( \cos^2 θ = (1 + \cos 2θ)/2 \), the integral becomes \( ½ a² [ 3θ/2 + 2 sin θ + sin 2θ/4 ]_0^{2π} = ½ a² (3π) = \frac{3π a²}{2} \).

利用 \( \cos^2 θ = (1 + \cos 2θ)/2 \)，积分变为 \( ½ a² [ 3θ/2 + 2 sin θ + sin 2θ/4 ]_0^{2π} = ½ a² (3π) = \frac{3π a²}{2} \)。

8. Rose Curves: Count the Petals | 玫瑰线：数清花瓣

Rose curves have equations like \( r = a \sin n\theta \) or \( r = a \cos n\theta \). The number of petals depends on whether \( n \) is even or odd:

玫瑰线的方程形如 \( r = a \sin n\theta \) 或 \( r = a \cos n\theta \)。花瓣数量取决于 \( n \) 是偶数还是奇数：

n	Number of petals (花瓣数)
odd (奇数)	n
even (偶数)	2n

For example, \( r = \sin 2\theta \) has 4 petals. Each petal is traced between successive zeros of \( r \): \( \theta = 0 \) to \( \pi/2 \) gives one petal. The area of one petal is:

例如，\( r = \sin 2\theta \) 有 4 片花瓣。每一片花瓣在 \( r \) 相继为零的角度之间描绘：\( \theta = 0 \) 到 \( \pi/2 \) 给出第一片花瓣。一片花瓣的面积为：

Area one petal = ½ ∫₀^π/2 (sin 2θ)² dθ = ½ ∫₀^π/2 sin² 2θ dθ

Using \( \sin^2 2θ = (1 – \cos 4θ)/2 \), we obtain \( ½ [θ/2 – sin 4θ/8]_0^{π/2} = ½ (π/4) = π/8 \). The total area is \( 4 × π/8 = π/2 \).

利用 \( \sin^2 2θ = (1 – \cos 4θ)/2 \)，得 \( ½ [θ/2 – sin 4θ/8]_0^{π/2} = ½ (π/4) = π/8 \)。总面积为 \( 4 × π/8 = π/2 \)。

9. Lemniscate Area | 双纽线面积

The lemniscate \( r^2 = a^2 \cos 2\theta \) consists of two loops. To find the area of one loop, set \( r = 0 \), giving \( \cos 2\theta = 0 \), so \( \theta = \pm \pi/4 \). For one loop, use \( -\pi/4 \) to \( \pi/4 \).

双纽线 \( r^2 = a^2 \cos 2\theta \) 由两片环组成。求一片环的面积时，令 \( r = 0 \)，得 \( \cos 2\theta = 0 \)，故 \( \theta = \pm \pi/4 \)。对于一片环，用 \( -\pi/4 \) 到 \( \pi/4 \)。

Area one loop = ½ ∫_-π/4^π/4 a² cos 2θ dθ = ½ a² [sin 2θ / 2]_-π/4^π/4 = ½ a² (1/2 + 1/2) = ½ a²

Therefore the total area of both loops is \( a^2 \).

因此两片环的总面积为 \( a^2 \)。

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

When the required region is bounded by two polar curves, integrate the difference of their squared radii. For example, find the area inside \( r = 3 \cos \theta \) but outside \( r = 1 + \cos \theta \).

当所求区域由两条极坐标曲线围成时，对其半径平方之差进行积分。例如，求在 \( r = 3 \cos \theta \) 内部但在 \( r = 1 + \cos \theta \) 外部的面积。

First find the intersection points: \( 3 \cos \theta = 1 + \cos \theta \), so \( 2 \cos \theta = 1 \), giving \( \theta = \pm \pi/3 \). The region is symmetric about the initial line, so compute from 0 to π/3 and double:

首先求交点：\( 3 \cos \theta = 1 + \cos \theta \)，得 \( 2 \cos \theta = 1 \)，因此 \( \theta = \pm \pi/3 \)。该区域关于极轴对称，因此计算 0 到 π/3 然后加倍：

Area = 2 × ½ ∫₀^π/3 [ (3 cos θ)² − (1 + cos θ)² ] dθ

Simplifying: \( 9\cos^2 θ – (1 + 2\cos θ + \cos^2 θ) = 8\cos^2 θ – 2\cos θ – 1 \). Integrate carefully to obtain the exact value.

化简：\( 9\cos^2 θ – (1 + 2\cos θ + \cos^2 θ) = 8\cos^2 θ – 2\cos θ – 1 \)。仔细积分得到精确值。

11. Common Mistakes and Tips | 常见错误与技巧

Forgetting the ½ factor in the area formula.

忘记面积公式中的 ½ 系数。
Using degrees instead of radians when integrating trigonometric functions.

积分三角函数时使用度数而不是弧度。
Integrating over the wrong interval, so that the curve is traced more than once or a negative radius is incorrectly used.

使用错误的积分区间，导致曲线被多次描绘，或错误地使用负半径。
When finding the area between two curves, ensure you subtract the inner squared radius from the outer squared radius; the integrand must be positive.

求两条曲线之间的面积时，务必用外半径的平方减去内半径的平方；被积函数必须为正。
Check for symmetry: many questions expect you to integrate over a small interval and multiply, which reduces the risk of sign errors.

检查对称性：许多题目期望你在小区间上积分然后乘以倍数，这能降低符号错误的风险。

12. Summary and Exam Strategy | 总结与应试策略

The area bounded by a polar curve always uses \( \frac{1}{2}\int r^2 \, d\theta \). Identify the limits from intersections with the pole or with another curve. Use symmetry to simplify the integral, and always verify that your final area is plausible for the given curve.

极坐标曲线所围面积始终使用 \( \frac{1}{2}\int r^2 \, d\theta \)。通过曲线与极点或另一条曲线的交点确定积分限；利用对称性简化积分；并始终检查最终面积是否与给定曲线相符。

In an AQA exam, show your working clearly, including the substitution of \( r = f(\theta) \) and the use of double-angle identities. State the limits in radians and write the exact answer in terms of \( \pi \) where possible.

在 AQA 考试中，要清晰地展示解题过程，包括代入 \( r = f(\theta) \) 和使用二倍角公式。以弧度为单位说明积分限，并尽可能用含 \( \pi \) 的精确值书写答案。

With systematic practice, polar area calculations become routine. Master the characteristic curves — circles, cardioids, roses, and lemniscates — and you will be ready for any AQA question.

通过系统的练习，极坐标面积计算会变得十分熟练。掌握典型曲线——圆、心形线、玫瑰线和双纽线——你就能从容应对任何 AQA 题目。

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The Area Bounded by a Polar Curve | 极坐标曲线所围面积