Arc Length and Area of Surface of Revolution | 弧长与旋转曲面面积

📚 Arc Length and Area of Surface of Revolution | 弧长与旋转曲面面积

When a curve is not a straight line, its total length cannot be measured with a ruler. Instead, we shrink the curve into infinitely many tiny straight segments, add their lengths by integration, and obtain a precise value. Exactly the same idea allows us to find the area of a surface formed when a curve is rotated about an axis.

当一条曲线不是直线时，我们不能用直尺直接量出它的总长度。我们可以把曲线切成无数个极小的直线段，用积分把它们加总起来，从而得到精确值。同样的思想也能帮助我们计算一条曲线绕某条轴旋转时所生成的曲面面积。

1. The Fundamental Building Block ds | 基本微分单元 ds

Consider a tiny segment of a curve. If the horizontal change is dx and the vertical change is dy, then the tiny length ds satisfies Pythagoras’ theorem, because the segment is locally almost straight.

考虑曲线上的一小段。若水平变化为 dx，垂直变化为 dy，则该微小长度 ds 满足勾股定理，因为在局部范围内曲线几乎可以看作直线。

ds² = dx² + dy²

This single relationship is the origin of every arc length and surface area formula in this article.

这个单一的关系式是本文所有弧长公式和旋转曲面面积公式的出发点。

2. Arc Length for Cartesian Curves y = f(x) | 直角坐标曲线 y = f(x) 的弧长

For a curve written as y = f(x), we can divide the relationship ds² = dx² + dy² by dx². This gives a formula for the small length ds in terms of the derivative dy/dx.

对于形如 y = f(x) 的曲线，我们将 ds² = dx² + dy² 两边除以 dx²，就能用导数 dy/dx 来表示微小长度 ds。

ds = √(1 + (dy/dx)²) dx

Integrating from x = a to x = b gives the total arc length L.

从 x = a 到 x = b 积分，即可得到总弧长 L。

L = ∫ₐᵇ √(1 + (dy/dx)²) dx

This formula is valid when the curve is smooth, meaning that dy/dx is continuous on the interval.

该公式仅在曲线光滑时成立，也就是要求 dy/dx 在区间上连续。

3. Worked Example 1: y = (2/3)x^(3/2) | 例 1：y = (2/3)x^(3/2)

Find the length of the curve y = (2/3)x^(3/2) from x = 0 to x = 3.

求曲线 y = (2/3)x^(3/2) 从 x = 0 到 x = 3 的弧长。

First differentiate with respect to x.

首先对 x 求导。

dy/dx = x^(1/2)

Substitute into the arc length formula.

代入弧长公式。

L = ∫₀³ √(1 + x) dx

Using the substitution u = 1 + x, or recognising the antiderivative directly, we get the exact value.

使用换元 u = 1 + x，或直接找出原函数，得到精确值。

L = [ (2/3)(1 + x)^(3/2) ]₀³ = (2/3)(8 − 1) = 14/3

Always check that the final answer is a positive number, and leave it in exact form if the question does not ask for a decimal.

始终检查最终答案为正数；若题目未要求小数，应保留精确形式。

4. Arc Length for Parametric Curves | 参数曲线的弧长

If a curve is given by parametric equations x = x(t) and y = y(t), then dx = (dx/dt) dt and dy = (dy/dt) dt. Substituting these into ds² = dx² + dy² gives the parametric arc length formula.

若曲线由参数方程 x = x(t) 与 y = y(t) 给出，则 dx = (dx/dt) dt，dy = (dy/dt) dt。把它们代入 ds² = dx² + dy²，即得到参数形式下的弧长公式。

L = ∫ from α to β √( (dx/dt)² + (dy/dt)² ) dt

Here α and β are the parameter values corresponding to the two ends of the curve. The formula does not require y to be a function of x; it works for closed curves and curves with vertical tangents too.

其中 α 和 β 是曲线两端对应的参数值。该公式不要求 y 必须是 x 的函数，也适用于封闭曲线或含有垂直切线的曲线。

5. Worked Example 2: x = t², y = t³ | 例 2：x = t², y = t³

Find the length of the curve x = t², y = t³ from t = 0 to t = 1.

求参数曲线 x = t²，y = t³ 从 t = 0 到 t = 1 的弧长。

Differentiate both coordinates with respect to t.

分别对参数 t 求导。

dx/dt = 2t, dy/dt = 3t²

Therefore the integrand becomes t√(4 + 9t²).

因此被积函数化为 t√(4 + 9t²)。

L = ∫₀¹ t√(4 + 9t²) dt

Using the substitution u = 4 + 9t², so du = 18t dt, gives the following result.

令 u = 4 + 9t²，则 du = 18t dt，得到如下结果。

L = (1/27)[ (4 + 9t²)^(3/2) ]₀¹ = (1/27)(13√13 − 8)

Notice how the factor t in the integrand is essential; it comes directly from the chain rule in the expression for ds.

注意被积函数中的因子 t 必不可少，它来自 ds 表达式中的链式法则。

6. Surface Area of Revolution about the x-axis | 绕 x 轴旋转的曲面面积

When a small segment of a curve is rotated about the x-axis, it sweeps out a band that is almost like a cylinder. The radius of this cylinder is approximately |y|, so its area is 2π|y| times the tiny length ds.

当曲线上的一小段绕 x 轴旋转时，它会扫出一个近似圆柱面的带状区域。该圆柱面的半径约为 |y|，因此其面积等于 2π|y| 乘以微小长度 ds。

Summing all such bands gives the surface area formula.

将所有带状面积相加，便得到曲面面积公式。

S = 2π ∫ₐᵇ |y| √(1 + (dy/dx)²) dx

For a parametric curve, replace ds by the parametric form.

对于参数曲线，将 ds 换成参数形式即可。

S = 2π ∫ from α to β |y(t)| √( (dx/dt)² + (dy/dt)² ) dt

The absolute value ensures the radius is never negative, which is especially important when the curve crosses the x-axis.

绝对值保证半径永远不为负，这在曲线穿过 x 轴时尤其重要。

7. Worked Example 3: Rotating a Straight Line | 例 3：旋转一条直线

The line y = 2x, for 0 ≤ x ≤ 1, is rotated about the x-axis. Find the area of the resulting surface.

直线 y = 2x，其中 0 ≤ x ≤ 1，绕 x 轴旋转。求所得曲面面积。

Here dy/dx = 2, so √(1 + (dy/dx)²) = √5.

这里 dy/dx = 2，所以 √(1 + (dy/dx)²) = √5。

S = 2π ∫₀¹ (2x)√5 dx = 4π√5 ∫₀¹ x dx

This evaluates to 2π√5, which matches the curved surface area of a cone with base radius 2 and slant height √5.

计算得 2π√5，这与底面半径为 2、母线长为 √5 的圆锥侧面积一致。

8. Surface Area of Revolution about the y-axis | 绕 y 轴旋转的曲面面积

If a curve is rotated about the y-axis, the radius of each tiny band is |x| rather than |y|.

若曲线绕 y 轴旋转，每个微小圆环带的半径是 |x| 而不是 |y|。

For a Cartesian curve x = g(y), the formula is written in terms of y.

对于直角坐标曲线 x = g(y)，公式以 y 为变量写出。

S = 2π ∫ from c to d |x| √(1 + (dx/dy)²) dy

For parametric curves, the formula becomes the following.

对于参数曲线，公式变为如下形式。

S = 2π ∫ from α to β |x(t)| √( (dx/dt)² + (dy/dt)² ) dt

Be careful: rotating about the y-axis does not simply swap x and y everywhere. The integration variable must match the coordinate along the axis of rotation.

请特别注意：绕 y 轴旋转并不是简单地把 x 和 y 互换。积分变量必须与旋转轴方向的坐标一致。

9. Polar Curves: An Extension | 极坐标曲线的推广

If your specification includes polar coordinates, the arc length of r = f(θ) from θ = α to θ = β is given by the formula below.

如果你的考试大纲包含极坐标，则曲线 r = f(θ) 从 θ = α 到 θ = β 的弧长公式如下。

L = ∫ from α to β √( r² + (dr/dθ)² ) dθ

The corresponding surface area of revolution about the initial line is obtained by using radius r sinθ.

绕极轴旋转的曲面面积则使用半径 r sinθ 来构造。

S = 2π ∫ from α to β r sinθ √( r² + (dr/dθ)² ) dθ

This is included here as a useful extension, particularly for AQA A-Level Further Mathematics candidates.

这里将其作为重要拓展列出，尤其适合 AQA 进阶数学的考生。

10. Common Mistakes and Exam Advice | 常见错误与考试建议

Forgetting to square both derivatives in the parametric formula. Every term must be squared before adding.

在参数公式中忘记给两个导数分别平方。每一项都必须先平方再相加。
Using the volume of revolution formula instead of the surface area formula. Surface area uses the factor 2π and includes ds, not dx alone.

误用旋转体体积公式代替曲面面积公式。曲面面积含有因子 2π，并且使用 ds 而不是单独的 dx。
Ignoring absolute values when the curve is below the axis. The radius of rotation cannot be negative.

当曲线位于 x 轴下方时忽略绝对值。旋转半径不可能为负。
Choosing the wrong limits. For parametric curves, limits are parameter values, not x- or y-coordinates.

选错积分上下限。对于参数曲线，上下限是参数值，而不是 x 坐标或 y 坐标。
Not simplifying the integrand before integrating. Algebra errors are the most common cause of lost marks.

积分前没有先化简被积函数。代数错误是失分最常见的原因。

11. Summary Table | 公式总结表

Curve Type	Arc Length	Surface Area about x-axis
y = f(x)	L = ∫ₐᵇ √(1 + (dy/dx)²) dx	S = 2π ∫ₐᵇ \|y\| √(1 + (dy/dx)²) dx
x = x(t), y = y(t)	L = ∫ from α to β √((dx/dt)² + (dy/dt)²) dt	S = 2π ∫ from α to β \|y(t)\| √((dx/dt)² + (dy/dt)²) dt
Polar r = f(θ)	L = ∫ from α to β √(r² + (dr/dθ)²) dθ	S = 2π ∫ from α to β r sinθ √(r² + (dr/dθ)²) dθ

Keep this table in mind as a single revision page. The central idea is always the same: build ds first, then integrate either ds itself for length, or 2π times a radius times ds for surface area.

请把这张表当作一页复习提纲。其核心思想始终一致：先构造 ds，再对 ds 本身积分求弧长，或对 2π 乘以半径再乘以 ds 积分求曲面面积。

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Arc Length and Area of Surface of Revolution | 弧长与旋转曲面面积