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, polar coordinates provide a powerful way to describe curves using a distance r from the pole and an angle θ from the initial line. One of the most common exam tasks is to calculate the area bounded by a polar curve, a region that is usually swept out by a rotating radius. This article derives the key formula, explains how to choose limits, and works through typical examples.

在 AQA A-Level 数学中，极坐标用点到极点的距离 r 和从极轴起算的角度 θ 来描述曲线。考试中最常见的任务之一是计算极坐标曲线围成的面积，这类区域通常由旋转的半径扫出。本文推导核心公式，说明如何选择积分限，并通过典型例题讲解。

1. Polar coordinates recap | 极坐标复习

A point P in polar coordinates is written as (r, θ), where r is the distance from the pole O and θ is the angle measured from the initial line, usually in radians. Unlike Cartesian coordinates, the same point can have infinitely many polar representations because adding multiples of 2π to θ gives the same direction.

极坐标中的点 P 写成 (r, θ)，其中 r 是到极点 O 的距离，θ 是从极轴开始测量的角，通常使用弧度。与直角坐标不同，同一个点可以有无穷多种极坐标表示，因为给 θ 加上 2π 的整数倍方向不变。

Many polar curves are given in the form r = f(θ). For example, r = a is a circle centred at the pole, and r = a(1 + cos θ) is a cardioid. When calculating areas, θ must be measured in radians because the sector area formula relies on radian measure.

许多极坐标曲线以 r = f(θ) 的形式给出。例如，r = a 是以极点为中心的圆，r = a(1 + cos θ) 是心形线。计算面积时，θ 必须使用弧度，因为扇形面积公式依赖弧度制。

2. Why area is not simply ∫ r dθ | 为什么面积不是简单 ∫ r dθ

A common first thought is to integrate r with respect to θ, but this does not give area. The reason is that the radius r changes as θ changes, and a thin sector of angle dθ has area approximately ½ r² dθ, not r dθ. Integrating r dθ would give the total change in radius, which is not related to area.

很多同学的第一反应是对 θ 积分 r，但这样得不到面积。原因是半径 r 随 θ 变化，角度为 dθ 的细小扇形面积近似为 ½ r² dθ，而不是 r dθ。对 r dθ 积分得到的是半径的总变化量，与面积无关。

Think of a circle of radius a. The area is πa², but integrating a with respect

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