Volume of Solids of Revolution by Integration | 旋转体体积的积分计算

📚 Volume of Solids of Revolution by Integration | 旋转体体积的积分计算

In IB Mathematics, the volume of a solid of revolution is one of the most important applications of definite integrals. When a region under a curve is rotated about a line, it sweeps out a three-dimensional solid, and we can calculate its volume exactly using integration. This article explains the core methods, formulas, and exam strategies you need to master this topic.

在IB数学中，旋转体的体积是定积分最重要的应用之一。当曲线下方的区域绕某条直线旋转时，会构成一个三维立体，我们可以通过积分精确计算其体积。本文将讲解核心方法、公式以及考试策略，帮助你完全掌握这一内容。

1. What is a Solid of Revolution? | 什么是旋转体？

A solid of revolution is created by taking a two-dimensional region bounded by curves and rotating it through 360° about a fixed axis. The axis of rotation can be the x-axis, the y-axis, or any horizontal or vertical line.

旋转体是由一个或几条曲线围成的二维区域，围绕固定轴旋转360°而形成的立体。旋转轴可以是x轴、y轴，也可以是任意水平或垂直直线。

Consider the region under the curve y = f(x) from x = a to x = b. If this region is rotated about the x-axis, the resulting solid has circular cross-sections perpendicular to the x-axis. The radius of each cross-section is exactly f(x), so the area of a cross-section is π[f(x)]².

考虑曲线 y = f(x) 在 x = a 到 x = b 之间的区域。如果将该区域绕x轴旋转，所得立体垂直于x轴的截面都是圆。每个截面的半径正好为 f(x)，因此截面积是 π[f(x)]²。

Integrating these cross-sectional areas along the axis of rotation gives the total volume.

将这些截面积沿旋转轴方向积分，便得到总体积。

2. The Disk Method (About the x-axis) | 圆盘法（绕x轴）

The disk method is used when the region under a curve is rotated about the x-axis. The volume is given by:

圆盘法适用于曲线下方区域绕x轴旋转的情形。其体积公式为：

V = ∫ₐᵇ π[f(x)]² dx

Here, f(x) represents the radius of the circular slice at position x. Each infinitesimal slice is treated as a disk of thickness dx.

其中 f(x) 表示位置 x 处圆形薄片的半径，每一个无穷小的薄片被看作厚度为 dx 的圆盘。

In IB problems, f(x) may be given explicitly, or the region may be defined by two curves. When two curves enclose a region and it is rotated about the x-axis, the cross-section is a ring, not a full disk.

在IB题目中，f(x) 可能直接给出，也可能区域由两条曲线围成。当两条曲线围成的区域绕x轴旋转时，截面是环形的，而不是完整的圆盘。

If the upper curve is y = f(x) and the lower curve is y = g(x), the outer radius is f(x) and the inner radius is g(x). 若上方曲线为 y = f(x)，下方曲线为 y = g(x)，则外半径为 f(x)，内半径为 g(x)。
The volume is found by subtracting the inner disk volume from the outer disk volume. 体积等于外圆盘体积减去内圆盘体积。

3. The Washer Method | 垫圈法（圆环法）

When the region between two curves is rotated about the x-axis, the cross-section perpendicular to the x-axis is a washer, a disk with a smaller disk removed from its centre. The formula is:

当两条曲线之间的区域绕x轴旋转时，垂直于x轴的截面是一个垫圈——一个圆盘中心挖去一个较小的圆盘。其公式为：

V = ∫ₐᵇ π( [f(x)]² − [g(x)]² ) dx

where f(x) is the outer radius and g(x) is the inner radius, with f(x) ≥ g(x) on [a, b].

其中 f(x) 是外半径，g(x) 是内半径，且在 [a, b] 上满足 f(x) ≥ g(x)。

Be careful: you must square each radius separately, then subtract. It is a common error to compute [f(x) − g(x)]² instead of [f(x)]² − [g(x)]².

注意：必须先分别平方，再相减。常见错误是写成 [f(x) − g(x)]²，而不是 [f(x)]² − [g(x)]²。

The washer method is also used when the region is above and below the x-axis, but the lower curve is taken as the inner radius.

当区域跨越x轴上下两侧时，垫圈法同样适用，只需将下方曲线作为内半径。

4. Rotation about the y-axis | 绕y轴旋转

If a region is rotated about the y-axis, the disk and washer methods are still valid, but the variables change. For a region bounded by x = h(y) and x = k(y), the volume is:

当区域绕y轴旋转时，圆盘法和垫圈法仍然适用，但变量需要改变。对于由 x = h(y) 和 x = k(y) 围成的区域，体积为：

V = ∫ₐᵇ π( [h(y)]² − [k(y)]² ) dy

In this case, the integration is performed with respect to y. The limits a and b are the y-coordinates of the bottom and top of the region.

此时积分变量变为 y，上下限 a 和 b 分别是区域最低点和最高点的 y 坐标。

Often the original curves are given as functions of x, such as y = x². To rotate about the y-axis, you must rearrange to x = √y, or use the shell method described below.

通常原始曲线是以 x 为自变量的函数，例如 y = x²。若要绕y轴旋转，需要将其改写为 x = √y，或者使用下面介绍的壳层法。

5. The Shell Method | 壳层法

The shell method (or cylindrical shells method) is an alternative technique, especially useful when rotating about the y-axis but the region is given as y = f(x). A thin vertical strip at position x, of height f(x) and width dx, sweeps out a cylindrical shell when rotated about the y-axis.

壳层法（又称圆柱壳法）是一种替代方法，特别适用于区域以 y = f(x) 形式给出但绕y轴旋转的情况。位于 x 处、高度为 f(x)、宽度为 dx 的竖直细条，绕y轴旋转后形成一个圆柱壳。

The volume of one shell is (circumference) × (height) × (thickness) = 2πx · f(x) · dx. Integrating gives:

一个壳层的体积为（周长）×（高度）×（厚度）= 2πx · f(x) · dx。积分得到：

V = ∫ₐᵇ 2πx f(x) dx

Here x is the distance from the strip to the y-axis (the radius of the shell), and f(x) is the height of the shell.

其中 x 是细条到y轴的距离（壳层的半径），f(x) 是壳层的高度。

Similarly, for rotation about the x-axis, the shell method uses horizontal strips with radius y and height expressed as a function of y.

类似地，绕x轴旋转时，壳层法使用水平细条，半径为 y，高度表示为 y 的函数。

6. Choosing Between Disk/Washer and Shell | 如何选择圆盘法/垫圈法与壳层法

The choice of method depends mainly on the axis of rotation and the form of the functions. If the region is described by y = f(x) and we rotate about the x-axis, the disk/washer method is usually simplest.

方法的选择主要取决于旋转轴和函数的形式。如果区域以 y = f(x) 表示并绕x轴旋转，那么圆盘法/垫圈法通常最简单。

Use washers when you can easily find an outer radius and an inner radius perpendicular to the axis. 当你容易找到垂直于旋转轴的外半径和内半径时，用垫圈法。
Use shells when the curves are already written as y = f(x) and you want to rotate about the y-axis, avoiding the need to rearrange. 当曲线已经写成 y = f(x) 而你要绕y轴旋转时，用壳层法，可以避免重新改写表达式。
In general, choose the method that leads to the simplest integrand. 一般来说，选择使被积函数最简单的方法。

Some volumes can be computed with either method, but the difficulty may differ significantly. On IB exams, always start by sketching the region and the axis of rotation.

有些体积用两种方法都可以计算，但难度可能差别很大。在IB考试中，一定要先画出区域和旋转轴的草图。

7. Rotating about Lines Other Than Axes | 绕非坐标轴旋转

Sometimes the rotation axis is not the x-axis or y-axis, but a horizontal line such as y = c or a vertical line such as x = d. The formulas must be adjusted by shifting the radii.

有时旋转轴不是x轴或y轴，而是水平线 y = c 或垂直线 x = d。此时需要平移半径来调整公式。

For rotation about y = c using washers, the radius of a horizontal slice is |f(x) − c|. The volume is:

对于绕 y = c 旋转用垫圈法，水平切片的半径为 |f(x) − c|。体积为：

V = ∫ₐᵇ π( [f(x) − c]² − [g(x) − c]² ) dx

Here f(x) and g(x) are the upper and lower curves, and c is the y-coordinate of the axis of rotation.

这里 f(x) 和 g(x) 分别是上曲线和下曲线，c 是旋转轴的 y 坐标。

For rotation about x = d using shells, the radius is |x − d|, so the volume becomes:

对于绕 x = d 旋转用壳层法，半径为 |x − d|，体积变为：

V = ∫ₐᵇ 2π |x − d| · f(x) dx

Take care with absolute values: if the region lies entirely on one side of the axis, you may drop the absolute value and adjust the sign accordingly.

注意绝对值的处理：如果区域完全位于旋转轴的一侧，你可以去掉绝对值并相应调整符号。

8. Worked Example 1: Disk Method | 例题1：圆盘法

Example. Find the volume of the solid obtained by rotating the region under y = √x from x = 0 to x = 4 about the x-axis.

例题。求由曲线 y = √x 下方从 x = 0 到 x = 4 的区域绕x轴旋转所得立体的体积。

Solution. Using the disk formula with f(x) = √x:

解。使用圆盘公式，f(x) = √x：

V = ∫₀⁴ π(√x)² dx = ∫₀⁴ π x dx

= π [x²/2]₀⁴ = π (16/2 − 0) = 8π

The volume is 8π cubic units.

体积为 8π 立方单位。

Notice that (√x)² = x, which simplifies the integrand greatly. Always simplify before integrating.

注意 (√x)² = x，这大大简化了被积函数。积分前一定要先化简。

9. Worked Example 2: Washer Method | 例题2：垫圈法

Example. Find the volume obtained by rotating the region bounded by y = x² and y = √x about the x-axis.

例题。求由曲线 y = x² 和 y = √x 围成的区域绕x轴旋转所得立体的体积。

Solution. First find the intersection points: x² = √x → x⁴ = x → x(x³ − 1) = 0, so x = 0 and x = 1. On [0, 1], √x ≥ x², so the outer radius is √x and the inner radius is x².

解。先求交点：x² = √x → x⁴ = x → x(x³ − 1) = 0，所以 x = 0 和 x = 1。在 [0, 1] 上，√x ≥ x²，因此外半径为 √x，内半径为 x²。

V = ∫₀¹ π( (√x)² − (x²)² ) dx = π ∫₀¹ (x − x⁴) dx

= π [x²/2 − x⁵/5]₀¹ = π (1/2 − 1/5) = 3π/10

The volume is 3π/10 cubic units.

体积为 3π/10 立方单位。

This example illustrates the importance of identifying the outer and inner radii correctly before integrating.

这个例子说明了在积分前正确识别外半径和内半径的重要性。

10. Worked Example 3: Shell Method | 例题3：壳层法

Example. The region under y = x² from x = 0 to x = 2 is rotated about the y-axis. Find the volume.

例题。曲线 y = x² 下方从 x = 0 到 x = 2 的区域绕y轴旋转，求体积。

Solution. Use the shell method because the curve is already solved for y in terms of x, and the rotation is about the y-axis.

解。使用壳层法，因为曲线已经写成 y 关于 x 的形式，且旋转轴是y轴。

V = ∫₀² 2πx · x² dx = 2π ∫₀² x³ dx = 2π [x⁴/4]₀²

= 2π (16/4) = 8π

The volume is 8π cubic units.

体积为 8π 立方单位。

If we attempted to use washers about the y-axis, we would need to solve y = x² for x as √y, and the limits would be y = 0 to y = 4. The integral becomes ∫₀⁴ π(√y)² dy = π ∫₀⁴ y dy = 8π, giving the same result.

如果我们尝试绕y轴用垫圈法，需要将 y = x² 解出 x = √y，上限为 y = 0 到 y = 4。积分变为 ∫₀⁴ π(√y)² dy = π ∫₀⁴ y dy = 8π，结果相同。

11. Common Pitfalls and Exam Tips | 常见误区与考试技巧

Do not forget the π in the disk and washer formulas. 不要忘记圆盘法和垫圈法公式中的 π。
For washers, always subtract the squares, not the radii. 对于垫圈法，一定要平方后再相减，而不是半径相减后再平方。
When using shells, the radius is the distance to the rotation axis, not the x-coordinate itself if the axis is shifted. 使用壳层法时，半径是到旋转轴的距离；如果轴有平移，则不是简单的 x 坐标。
Always determine the limits of integration by solving for intersection points or reading the given bounds. 始终通过解交点或读取给定的边界来确定积分上下限。
Sketch the region and axis first; this prevents sign errors and wrong radii. 先画出区域和旋转轴的草图；这可以防止符号错误和半径错误。
Check whether the region is above/below the axis. If part of the region lies below the x-axis, the radius in the disk method must use the absolute value or a different setup. 检查区域在轴的上方还是下方。如果区域有一部分在x轴下方，圆盘法中的半径必须使用绝对值或采用不同设置。

On the IB exam, show your integration steps clearly and write the final answer with units. Methods are often awarded credit even if the final number is wrong.

在IB考试中，要清晰地写出积分步骤，并在最终答案中加上单位。即使最终数字错误，正确的解题方法也往往能获得部分分数。

12. Practice Questions | 练习

Try these problems to test your understanding:

尝试以下问题来测试你的理解：

Find the volume of the solid formed by rotating the region under y = x³ from x = 0 to x = 1 about the x-axis. 求曲线 y = x³ 下方从 x = 0 到 x = 1 的区域绕x轴旋转所得立体的体积。
Find the volume when the region between y = x and y = x² is rotated about the x-axis. 求由 y = x 和 y = x² 围成的区域绕x轴旋转所得立体的体积。
The region bounded by y = eˣ, the x-axis, x = 0 and x = 1 is rotated about the y-axis. Use the shell method to set up the integral. 由 y = eˣ、x轴、x = 0 和 x = 1 围成的区域绕y轴旋转。用壳层法建立积分表达式。

13. Summary | 总结

The volume of a solid of revolution can be found by integrating cross-sectional areas or cylindrical shells. The disk method uses π[f(x)]², the washer method uses π([f(x)]² − [g(x)]²), and the shell method uses 2πx·f(x). Choose the method that matches the axis of rotation and the given function form.

旋转体的体积可以通过积分截面面积或圆柱壳层来求得。圆盘法使用 π[f(x)]²，垫圈法使用 π([f(x)]² − [g(x)]²)，壳层法使用 2πx·f(x)。根据旋转轴和给定的函数形式选择合适的方法。

Mastering these formulas and understanding when to apply them is essential for success in IB Mathematics. Always practise with a sketch, and verify your results using a different method when possible.

掌握这些公式并理解何时应用它们，是IB数学取得好成绩的关键。练习时始终画草图，并尽可能用另一种方法验证你的结果。

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Volume of Solids of Revolution by Integration | 旋转体体积的积分计算