Solids of Revolution | 旋转体

📚 Solids of Revolution | 旋转体

In integral calculus, a solid of revolution is formed by rotating a plane region about a given line (the axis of revolution). This concept leads to practical applications in computing volumes of objects such as vases, bowls, and mechanical parts. The volume is typically found using disk, washer, or shell methods, all derived from summing infinitesimally thin cross-sectional areas. Understanding these methods deepens your insight into how integration models real-world shapes.

在积分学中，旋转体是由平面区域绕某一直线（旋转轴）旋转一周所形成的立体图形。这一概念可用于计算花瓶、碗、机械零件等物体的体积，具有广泛应用。体积通常采用圆盘法、垫圈法或壳层法求得，这些方法均基于对无穷薄截面面积的累加。掌握这些方法能帮助你深刻理解积分如何描述实际形状。

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

A solid of revolution is a three-dimensional solid obtained by rotating a two-dimensional region around an axis. The axis can be horizontal (like the x-axis) or vertical (like the y-axis). When the region is rotated, each point traces a circle, and the union of these circles forms the solid. For example, rotating a rectangle around one of its edges generates a cylinder; rotating a right triangle around a leg produces a cone.

旋转体是将一个二维区域围绕某轴旋转得到的立体图形。该轴可以是水平的（如 x 轴）或竖直的（如 y 轴）。当区域旋转时，每一个点描绘出一个圆，这些圆的并集形成整个立体。例如，矩形绕其一条边旋转生成圆柱；直角三角形绕一直角边旋转生成圆锥。

Cross sections perpendicular to the axis of revolution are either solid disks (if the region touches the axis) or washers (if there is a gap). The integration strategy simply adds up the volumes of infinitely many such cross-sectional slices. In IB Mathematics analysis, you will be expected to set up and evaluate these volume integrals using definite integration techniques.

垂直于旋转轴的截面要么是实心圆盘（若区域接触轴线），要么是圆环/垫圈（若存在空隙）。积分的策略就是将无穷多个这种切片的体积累加起来。在 IB 数学分析课程中，要求你能够建立并计算这类体积积分，运用定积分技巧求解。

2. Volume by Disk Method (About the x-axis) | 圆盘法求体积（绕 x 轴）

If a plane region is bounded by the curve y = f(x) ≥ 0, the x-axis, and the vertical lines x = a and x = b, rotating it around the x-axis creates a solid composed of circular disks. At a typical point x, the radius of the disk is simply R(x) = f(x), assuming f(x) is non‑negative. The area of that cross‑sectional disk is π [R(x)]². Multiplying by an infinitesimal thickness Δx gives the volume of a slice, and integrating yields the total volume.

若平面区域由曲线 y = f(x) ≥ 0、x 轴以及垂直线 x = a 与 x = b 围成，绕 x 轴旋转即得由一系列圆盘组成的立体。在 x 处，圆盘半径就是 R(x) = f(x)，假设 f(x) 非负。该截面圆盘的面积为 π [R(x)]²。乘以微元厚度 Δx 即得切片体积，再积分即得总体积。

V = π ∫_a^b [f(x)]² dx

When f(x) takes negative values you must use |f(x)| for the radius, or restrict the domain to the part where f(x) ≥ 0 and use symmetry if the region is symmetric. The formula assumes the region touches the axis, so no inner hole appears.

当 f(x) 取负值时，半径应取 |f(x)|，或将定义域限制在 f(x) ≥ 0 的区间，若区域对称则可利用对称性处理。该公式假定区域紧贴轴线，因此不产生内孔。

3. Volume by Disk Method (About the y-axis) | 圆盘法求体积（绕 y 轴）

Revolving a region around the y-axis requires expressing the boundary curve in the form x = g(y), where g(y) ≥ 0 on the interval y = c to y = d. The radius of a representative disk is R(y) = g(y), and the thickness is Δy. Summing the slices gives a volume integral with respect to y. This approach is particularly useful when the region is naturally described by functions of y.

绕 y 轴旋转需要将边界曲线写成 x = g(y) 的形式，且在区间 y = c 到 y = d 上 g(y) ≥ 0。代表圆盘的半径为 R(y) = g(y)，厚度为 Δy。对切片求和得到关于 y 的体积积分。当区域用 y 的函数描述起来更方便时，此方法尤为实用。

V = π ∫_c^d [g(y)]² dy

Always check that you have correctly inverted the original function. If the original curve is y = f(x), find the inverse function x = f⁻¹(y) or, if the curve is not one‑to‑one, split the region into monotonic branches. For IB exams, curves are typically chosen so that inversion is straightforward, for example y = x² → x = √y for the right branch.

务必确保正确求反函数。若原曲线为 y = f(x)，需找出反函数 x = f⁻¹(y)；若函数不是一一对应，可将区域分割为单调分支。IB 考试通常选取容易求反函数的曲线，例如 y = x² → 右侧分支 x = √y。

4. The Washer Method for Hollow Solids | 垫圈法（圆环法）求空心旋转体

When the region to be revolved does not touch the axis of rotation, or when the solid has a cavity, the cross‑section perpendicular to the axis is a washer – a disk with a hole. Suppose the region is bounded between two curves y = f(x) and y = g(x) with f(x) ≥ g(x) ≥ 0 on [a, b], and it is revolved about the x‑axis. The outer radius is R(x) = f(x) and the inner radius is r(x) = g(x). The area of the washer is π (R² – r²). The volume is then obtained by integrating this area along the axis.

当旋转区域不接触旋转轴，或立体内部存在空腔时，垂直于轴的截面呈圆环状——即中心有孔的圆盘。假设区域介于两条曲线 y = f(x) 和 y = g(x) 之间，在 [a, b] 上满足 f(x) ≥ g(x) ≥ 0，并绕 x 轴旋转。外半径 R(x) = f(x)，内半径 r(x) = g(x)。圆环面积为 π (R² – r²)。体积即沿轴向对该面积积分所得。

V = π ∫_a^b ( [f(x)]² – [g(x)]² ) dx

The washer method also applies when one of the boundaries is the axis itself and the other is a curve not touching the axis – then the inner radius is zero and it reduces to the disk method. It is essential to identify the larger and smaller radii correctly; the larger function gives the outer boundary.

垫圈法也适用于边界之一是轴线、另一边界是不接触轴线的曲线的情形——此时内半径为零，公式退化为圆盘法。关键是要正确识别较大的半径（外边界）和较小的半径（内边界），较大的函数值对应外边界。

5. Revolving About Horizontal Lines y = c | 绕水平线 y = c 旋转

If the axis of revolution is shifted to a horizontal line y = c (not the x‑axis), the radius at a given x is the vertical distance from the curve to the line. For a region above the line bounded by y = f(x) and y = c, the outer radius becomes f(x) – c. When using the washer method, both boundaries must be adjusted by subtracting c. The integral bounds remain along the x‑axis.Published by TutorHao | IB Mathematics Revision Series | aleveler.com

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