Volume of a Torus by Definite Integral | 用定积分推导圆环体体积

📚 Volume of a Torus by Definite Integral | 用定积分推导圆环体体积

A torus is a three-dimensional shape that resembles a doughnut, formed by rotating a circle around an axis that lies in the same plane as the circle but does not intersect it. In this article, we will use the power of definite integration to derive its volume step by step.

圆环体是一种类似甜甜圈的三维形状，它是由一个圆绕着与之共面但不相交的轴旋转一周而形成的。本文将利用定积分的力量，一步步推导它的体积公式。

1. What Is a Torus? | 什么是圆环体？

Consider a circle of radius r whose centre is at a distance R from a vertical axis. If this circle is rotated around the axis, it sweeps out a torus. The distance R is called the major radius, and r is the minor radius.

考虑一个半径为 r 的圆，其圆心到一条竖直轴的距离为 R。当这个圆绕该轴旋转一周时，它扫过的立体就是圆环体。距离 R 称为主半径，而 r 称为次半径。

For a genuine torus, we require R > r so that the shape has a hole in the middle. The volume depends only on these two parameters.

要形成一个真正的圆环体，需要满足 R > r，这样中间才会有一个孔。其体积仅由这两个参数决定。

2. Why Use Integration? | 为什么用定积分？

Integration is the natural tool for computing volumes of solids whose cross-sectional areas vary continuously. The volume of a solid can be obtained by integrating its cross-sectional area along a chosen axis.

积分是计算截面面积连续变化的立体体积的天然工具。一个立体的体积可以通过沿某一坐标轴对其截面积进行积分来得到。

For a torus, each horizontal slice is a flat annulus (a “washer”). The area of this washer changes with the height of the slice, which makes a definite integral perfectly suited to the problem.

对于圆环体，每一个水平切片都是扁平的圆环（即“垫圈”）。这个圆环的面积随着切片的高度而变化，因此用定积分来处理这一问题非常合适。

3. The Slicing Method | 切片法（垫圈法）

We place the torus so that its axis of rotation is the z-axis. A horizontal plane at height z cuts the torus in a circular ring. We first find the area A(z) of that ring, and then sum all such slices using an integral.

我们将圆环体的旋转轴放在 z 轴上。高度为 z 的水平面会把圆环体截成一个圆环。我们首先求出该圆环的面积 A(z)，然后通过积分将所有这样的切片累加起来。

If the plane does not intersect the torus, A(z) is zero. The slicing only occurs between z = −r and z = r, because the original circle extends only r units above and below its centre.

如果平面不与圆环体相交，则 A(z) 为零。切片只发生在 z = −r 到 z = r 之间，因为原始圆只在中心上下各延伸 r 个单位。

4. Setting Up Coordinates | 建立坐标系

Let the original circle lie in the xz-plane, with its centre at (R, 0, 0). Its equation is:

设原始圆位于 xz 平面内，圆心为 (R, 0, 0)。其方程为：

(x − R)² + z² = r²

At a fixed height z, the value of x on the circle satisfies x = R ± √(r² − z²). Because R > r, both of these values are positive.

在固定高度 z 处，圆上的 x 值满足 x = R ± √(r² − z²)。由于 R > r，两个值均为正数。

When this point is rotated around the z-axis, the distance from the z-axis to the surface is |x| = x. Therefore, the inner radius of the washer is R − √(r² − z²), and its outer radius is R + √(r² − z²).

当该点绕 z 轴旋转时，从 z 轴到表面的距离为 |x| = x。因此，垫圈的内半径为 R − √(r² − z²)，外半径为 R + √(r² − z²)。

5. Area of a Cross-Section | 横截面面积

The area of a washer is the difference between the areas of two circles:

一个垫圈的面积是两个圆面积的差：

A(z) = π[(R + √(r² − z²))² − (R − √(r² − z²))²]

Expand the squares. The R² terms cancel, and the cross terms add:

展开平方项。R² 项相互抵消，交叉项相加：

A(z) = π[4R√(r² − z²)] = 4πR√(r² − z²)

Notice that the area is proportional to √(r² − z²), which is largest at z = 0 and vanishes at z = ±r.

注意面积与 √(r² − z²) 成正比，在 z = 0 时最大，在 z = ±r 时变为零。

6. Integrating to Get Volume | 积分求体积

The volume V is the integral of A(z) from z = −r to z = r:

体积 V 是 A(z) 从 z = −r 到 z = r 的积分：

V = ∫_−r^r A(z) dz = ∫_−r^r 4πR√(r² − z²) dz

Because 4πR is a constant, it can be pulled out of the integral:

因为 4πR 是常数，可以提到积分号外：

V = 4πR ∫_−r^r √(r² − z²) dz

This remaining integral is exactly the area of a semicircle of radius r, multiplied by 2. In other words, ∫_−r^r √(r² − z²) dz equals half the area of a full circle, which is πr²/2.

剩下的这个积分恰好是半径为 r 的半圆面积的两倍。换句话说，∫_−r^r √(r² − z²) dz 等于整个圆面积的一半，即 πr²/2。

7. Evaluating the Integral | 计算积分

Using the geometric formula, we do not need to perform a trigonometric substitution. The exact value is:

利用几何公式，我们无需进行三角换元。其精确值为：

∫_−r^r √(r² − z²) dz = ½πr²

Substitute this back into the expression for the volume:

将这一结果代回体积表达式中：

V = 4πR × ½πr² = 2π²Rr²

Thus the volume of the torus is 2π²Rr². This is the famous torus volume formula obtained through definite integration.

因此，圆环体的体积为 2π²Rr²。这正是通过定积分得到的著名圆环体体积公式。

8. Final Formula | 最终公式

For a torus with major radius R and minor radius r, where R > r:

对于主半径为 R、次半径为 r（R > r）的圆环体：

V = 2π²Rr²

This formula is remarkably simple and depends linearly on R but quadratically on r.

这个公式非常简洁，它关于 R 是线性的，而关于 r 是二次的。

9. Example: A Standard Torus | 示例：标准圆环体

Take a torus with R = 5 cm and r = 2 cm. Its volume is:

取一个圆环体，其中 R = 5 cm，r = 2 cm。其体积为：

V = 2π² × 5 × 2² = 2π² × 5 × 4 = 40π² cm³ ≈ 394.8 cm³

If we double the minor radius to r = 4 cm while keeping R the same, the volume becomes 2π² × 5 × 16 = 160π² cm³, which is four times as large. This shows the quadratic dependence on r.

如果保持 R 不变，将次半径加倍为 r = 4 cm，体积变为 2π² × 5 × 16 = 160π² cm³，是原来的四倍。这体现了体积对 r 的二次依赖关系。

10. Connection to Pappus’s Theorem | 与帕普斯定理的联系

Pappus’s centroid theorem states that the volume of a solid of revolution equals the area of the generating plane figure multiplied by the distance travelled by its centroid. For the torus, the generating circle has area πr² and its centroid travels a distance 2πR. Thus:

帕普斯重心定理指出，旋转体的体积等于生成平面图形的面积乘以其重心经过的距离。对于圆环体，生成圆的面积为 πr²，其重心经过的距离为 2πR。因此：

V = πr² × 2πR = 2π²Rr²

Our integration result matches Pappus’s theorem perfectly. This cross-check confirms the correctness of the definite-integral approach.

我们的积分结果与帕普斯定理完全一致。这种交叉验证证实了定积分方法的正确性。

11. Common Mistakes | 常见错误

Students often make the following errors when deriving or using the torus volume formula.

学生在推导或使用圆环体体积公式时常犯以下错误。

Confusing R and r: R is the distance from the axis to the centre of the generating circle, while r is the radius of that circle. Swapping them gives a wrong formula.

混淆 R 和 r：R 是旋转轴到生成圆心的距离，而 r 是该圆的半径。将两者颠倒会得到错误的公式。
Forgetting the factor 2π: Some use the area of the circle once instead of the full rotational distance. The factor 2π is essential.

忘记系数 2π：有人只乘了圆的面积而忽略了旋转一周的距离。系数 2π 是必不可少的。
Incorrect cross-section area: Using (R + √)² − (R − √)² incorrectly may occur when the expression is expanded. Remember the cross terms add to 4R√.

截面面积计算错误：在展开 (R + √)² − (R − √)² 时容易出错。注意交叉项相加为 4R√。
Limits of integration: The integral must run from −r to r, not from −R to R. The height of the torus is determined by the minor radius r.

积分上下限错误：积分必须从 −r 到 r，而不是从 −R 到 R。圆环体的高度由次半径 r 决定。

12. Summary | 总结

By slicing the torus perpendicular to its axis, we expressed the cross-sectional area as A(z) = 4πR√(r² − z²). Integrating this area from −r to r produced the volume formula:

通过垂直于旋转轴切片，我们得到横截面积 A(z) = 4πR√(r² − z²)。将该面积从 −r 到 r 积分，得到体积公式：

V = 2π²Rr²

This elegant result shows how a simple double-sweep in space can be captured by a single definite integral. Mastering this technique builds a strong foundation for solving more advanced volume problems in rotation geometry.

这一优雅的结果表明，空间中的一次双重扫掠可以通过一个定积分简洁地刻划。掌握这一技巧，为解决旋转几何中更复杂的体积问题奠定了坚实的基础。

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Volume of a Torus by Definite Integral | 用定积分推导圆环体体积