Area of a Surface of Revolution | 旋转曲面的面积

📚 Area of a Surface of Revolution | 旋转曲面的面积

A surface of revolution is formed when a curve rotates through 360 degrees around a fixed axis. In AQA Further Mathematics, you need to be able to use integration to find the curved surface area generated by a curve rotating about the x-axis or the y-axis.

旋转曲面是由一条曲线绕固定轴旋转 360 度后形成的曲面。在 AQA 进阶数学中，你需要掌握用积分计算曲线绕 x 轴或 y 轴旋转所得曲面的面积。

1. What is a Surface of Revolution? | 什么是旋转曲面

A surface of revolution is created when a curve is rotated through a full turn around a straight line, usually the x-axis or the y-axis. The original curve is called the generator, and the fixed line is called the axis of rotation.

旋转曲面是将一条曲线绕一条直线旋转一整周而形成的曲面，这条直线通常是 x 轴或 y 轴。原来的曲线称为母线，固定的直线称为旋转轴。

Common examples include spheres, cones, paraboloids, and vase-shaped surfaces. In exam questions, you are usually asked to find the area of the curved surface only, not the volume inside it.

常见的例子包括球面、圆锥面、抛物面和花瓶状曲面。考试中通常只要求你求出曲面的面积，而不是其内部体积。

It is useful to visualise the curve as a piece of string. When the string is spun around the axis, it sweeps out a thin band whose surface area can be added up by integration.

把曲线想象成一根细绳会很有帮助。当这根细绳绕轴旋转时，它会扫出一个细带，所有这些细带的面积可以通过积分累加。

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2. The Core Formula: Rotation About the x-Axis | 核心公式：绕 x 轴旋转

d about the x-axis from x = a to x = b, the curved surface area S is given by the formula:

如果曲线 y = f(x) 绕 x 轴旋转，x 的范围是从 a 到 b，则曲面面积 S 由以下公式给出：

S = 2π ∫_a^b y √(1 + (dy/dx)²) dx

is the small arc length ds. Therefore, 2π y ds is the area of a thin band around the axis.

在这个公式中，y 是 x 处圆形截面的半径。表达式 √(1 + (dy/dx)²) dx 是微元弧长 ds。因此，2π y ds 就是绕轴的一个细带面积。

Note that the formula gives the lateral surface area only. It does not include the areas of any circles at the ends of the solid unless the question specifically asks for them.

请注意，这个公式只给出侧面积。除非题目特别要求，否则它不包括立体两端的圆形面积。

• If the radius y is measured from the x-axis, it must be non-negative. If the curve crosses the axis, split the integral at the crossing points.
• 如果半径 y 是从 x 轴测量的，它必须是非负的。如果曲线穿过 x 轴，就要在交点处把积分分开。

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3. Rotation About the y-Axis | 绕 y 轴旋转

the formula changes because the radius is now x, the horizontal distance from the y-axis.

当曲线 x = g(y) 绕 y 轴旋转，y 从 c 到 d 时，公式会发生变化，因为此时半径是 x，也就是到 y 轴的水平距离。

S = 2π ∫_c^d x √(1 + (dx/dy)²) dy

ngth element is √(1 + (dx/dy)²) dy because the variable of integration is y rather than x.

这里的弧长元素是 √(1 + (dx/dy)²) dy，因为积分变量是 y 而不是 x。

A very common exam error is to use y as the radius when the rotation is about the y-axis. Always remind yourself: the radius is the perpendicular distance from the curve to the axis of rotation.

一个非常常见的考试错误是：绕 y 轴旋转时仍然用 y 作为半径。一定要提醒自己：半径是曲线到旋转轴的垂直距离。

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4. Parametric Curves | 参数方程形式

he surface area can be found by replacing the arc length element with the parametric form.

如果曲线由参数方程 x = x(t), y = y(t) 给出，且 t 的范围是从 α 到 β，那么可以把弧长元素替换为参数形式来求曲面面积。

Rotation about the x-axis:

绕 x 轴旋转：

S = 2π ∫_α^β y(t) √((dx/dt)² + (dy/dt)²) dt

Rotation about the y-axis:

绕 y 轴旋转：

S = 2π ∫_α^β x(t) √((dx/dt)² + (dy/dt)²) dt

These forms are especially useful when y cannot be written easily as a function of x, or when the integral is simpler in the parameter t.

当 y 不容易写成 x 的显式函数，或者积分在参数 t 下更简单时，这些参数形式尤其有用。

Axis	Surface area formula
x-axis, y = f(x)	S = 2π ∫ab y √(1 + (dy/dx)²) dx
y-axis, x = g(y)	S = 2π ∫cd x √(1 + (dx/dy)²) dy
x-axis, parametric	S = 2π ∫αβ y(t) √((dx/dt)² + (dy/dt)²) dt
y-axis, parametric	S = 2π ∫αβ x(t) √((dx/dt)² + (dy/dt)²) dt

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5. Where the Formula Comes From: Frustum Idea | 公式从何而来：圆台思想

To understand why the formula works, split the curve into many tiny pieces. Each piece has length ds. When one piece rotates about the x-axis, it sweeps out a thin band that looks like the side of a frustum of a cone.

要理解这个公式为什么成立，可以把曲线分成许多小段。每一段的长度为 ds。当其中一小段绕 x 轴旋转时，它会扫出一个细带，形状像一个圆台的侧面。

The lateral area of a frustum is approximately 2π times the average radius times the slant height. Since the slice is very small, the radius can be taken as y, and the slant height is ds. Thus the small area is 2π y ds.

圆台的侧面积约为 2π 乘以平均半径再乘以斜高。由于小段非常短，半径可以近似取为 y，斜高为 ds。因此小面积为 2π y ds。

as ds = √(1 + (dy/dx)²) dx. Integrating all the small bands from a to b gives the formula.

根据毕达哥拉斯定理，ds² = dx² + dy²。如果 y 是 x 的函数，这可以写成 ds = √(1 + (dy/dx)²) dx。将所有细带面积从 a 到 b 积分，就得到公式。

This explains why you must never replace ds with just dx. The arc length includes both horizontal and vertical changes, so the square root factor is essential.

这就解释了为什么不能把 ds 直接替换成 dx。弧长同时包含水平变化和垂直变化，因此平方根因子是必不可少的。

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6. Worked Example 1: Spherical Cap | 例题1：球冠面积

Find the surface area generated when the curve y = √4 − x², from x = 0 to x = 1, is rotated about the x-axis.

求曲线 y = √4 − x² 从 x = 0 到 x = 1 绕 x 轴旋转所得曲面的面积。

First find the derivative. Since y = √4 − x², we have:

先求导数。因为 y = √(

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