📚 Areas and Volumes (Integration Applications) | 积分应用：面积与体积
Definite integrals allow us to measure not just the area under a curve but also the volume generated when a region is rotated about a line. In the IB Mathematics: Analysis and Approaches (AA) course, both Standard and Higher Level students are expected to apply integration techniques to find areas bounded by curves and volumes of revolution. This article covers the essential formulas, methods, and common pitfalls, providing a thorough revision of 'C Areas and volumes' from the calculus syllabus.
定积分不仅能计算曲线下方的面积，还能求出将平面区域绕某条直线旋转后所生成的体积。在 IB 数学：分析与方法（AA）课程中，无论是标准水平还是高水平学生，都需要运用积分技巧计算曲线围成的面积和旋转体体积。本文全面覆盖微积分部分“面积与体积”的核心公式、方法与常见错误，帮助同学们系统复习。
1. Definite Integrals: The Key Concept | 定积分：核心概念
Areas and volumes are calculated using the definite integral, which represents the limit of a Riemann sum. For a continuous function f(x), the area between the graph and the x-axis from x = a to x = b is given by ∫ab f(x) dx, provided f(x is, the function dips below the axis, the integral yields a signed value, so absolute values or splitting the interval become necessary to find the true geometric area. Similarly, volumes of revolution are built up by summing thin slices (disks or washers) whose cross‑sectional areas are integrated.
面积和体积通过定积分计算，定积分是黎曼和的极限。对于连续函数 f(x)，当 f(x) ≥ 0 时，从 x = a 到 x = b 的图像与 x 轴之间的面积为 ∫ab f(x) dx。如果函数部分在 x 轴下方，积分给出的是有正负的面积，此时需要取绝对值或拆分区间才能得到真实的几何面积。旋转体体积则通过累加薄片（圆盘或垫圈）的横截面积来求得。
2. Area Between a Curve and the x-axis | 曲线与x轴之间的面积
To compute the area of the region bounded by y = f(x), the x-axis, and the vertical lines x = a and x = b, first determine where f(x) changes sign on [a, b]. If f(x) is positive throughout, the area is simply A = ∫ab f(x) dx crosses the axis, split the integral at the roots and take the absolute value of each sub‑integral: A = ∫ac |f(x)| dx + ∫cb |f(x)| dx, where c is the intercept. Never blindly integrate without checking the sign; doing so can lead to negative 'areas' and an incorrect final answer.
计算由 y = f(x)、x 轴以及直线 x = a、x = b 围成的区域面积时，首先要判断 f(x) 在 [a, b] 上是否变号。如果函数始终非负，面积即为 A = ∫ab f(x) dx。若图像穿越 x 轴，就需要在零点处拆分积分，并将每段积分取绝对值：A = ∫ac |f(x)| dx + ∫cb |f(x)| dx，其中 c 为零点。切勿在不判断正负的情况下直接积分，否则可能出现“面积为负”的荒谬结果。
3. Area Between Two Curves | 两条曲线之间的面积
If a region is bounded by y = f(x) (top) and y = g(x) (bottom) between x = a and x = b, the area is A = ∫ab [f(x) − g(x)] dx. The formula works regardless of whether the curves lie above the x-axis, as long as f(x) ≥ g(x urves intersect, find the intersection points and integrate the absolute difference: A = ∫ab |f(x) − g(x)| dx. Many IB questions require you first to find the limits from the intersection points, and sometimes to set up the integral with respect to y rather than x when it is more convenient.
若区域由 y = f(x)（上曲线）和 y = g(x)（下曲线）以及 x = a 和 x = b 围成，则面积为 A = ∫ab [f(x) − g(x)] dx。只要在区间上 f(x) ≥ g(x)，该公式就成立，与曲线是否在 x 轴上方无关。如果曲线相交，先求出交点，再对绝对值差积分：A = ∫ab |f(x) − g(x)| dx。许多 IB 考题会要求先通过解方程找到积分限，有时还会发现对 y 积分比用 x 更方便。
4. Areas Involving the y-axis | 以y轴为边界的面积
When a region is better described by x = g(y), area calculations can be performed along the y-axis. For instance, the area bounded by the curve x = g(y), the y-axis, and horizontal lines y = c and y = d is A = ∫cd g(y) dy (assuming g(y) ≥ 0). The area between two curves x = gright(y) and x = gleft(y) from y = c to d is A = ∫cd [gright(y) − gleft(y)] dy. This approach is particularly useful for functions like y = ln x, where converting to x = ey makes the integral simpler.
当区域更适合由 x = g(y) 描述时，我们可以沿 y 轴计算面积。例如，曲线 x = g(y)、y 轴以及水平线 y = c、y = d 围成的面积为 A = ∫cd g(y) dy（假设 g(y) ≥ 0）。而两条曲线 x = gright(y) 与 x = gleft(y) 在 y = c 至 d 之间的面积则为 A = ∫cd [gright(y) − gleft(y)] dy。当遇到 y = ln x 这样的函数时，改写成 x = ey 后积分会简单得多，这种换位思路在考试中十分实用。
5. Volumes of Revolution: Introduction | 旋转体体积导论
A solid of revolution is formed by rotating a plane region through 360° about a line (usually the x-axis or y-axis). The volume can be found by the disk method: slicing the solid perpendicular to the axis of rotation. Each slice is a thin disk of radius r and thickness dx (or dy), and its volume is πr² dx. Integrating these infinitesimal volumes gives the total volume. When there is a hole in the solid – for example, rotating the region between two curves – the washer method subtracts the inner disk volume from the outer disk volume.
将平面区域绕某条直线（通常是 x 轴或 y 轴）旋转 360° 即可生成一个旋转体。体积可用圆盘法求得：垂直于旋转轴将立体切成薄片，每片都是一个半径为 r、厚度为 dx（或 dy）的薄圆盘，其体积为 πr² dx。对这些微小体积积分就得到总体积。如果旋转体内部有空洞（例如旋转两条曲线之间的区域），则使用垫圈法，用外圆盘体积减去内圆盘体积。
6. Volume of Revolution about the x-axis (Disk Method) | 绕x轴旋转体积（圆盘法）
If the region bounded by y = f(x), the x-axis, x = a, and x = b is rotated 360° about the x-axis, the resulting volume is
V = π ∫ab [f(x)]² dx
This formula assumes the region lies entirely on one side of the axis. The radius of each disk is simply the y-coordinate of the curve, so the cross‑sectional area is π[f(x)]². If the region extends below the x-axis, the square eliminates any sign concerns, but you must still define the region clearly – often by considering the absolute value or taking the top curve as the outer radius.
若将由 y = f(x)、x 轴以及直线 x = a、x = b 围成的区域绕 x 轴旋转 360°，所得体积为
V = π ∫ab [f(x)]² dx
该公式要求区域完全位于轴的一侧。每个圆盘的半径即为曲线的 y 坐标，因此横截面积为 π[f(x)]²。如果区域部分在 x 轴下方，平方会自动消除正负问题，但务必明确区域的边界——通常需要取绝对值，或将上方曲线作为外半径。
7. Volume of Revolution about the y-axis | 绕y轴旋转体积
When a region bounded by x = g(y), the y-axis, and the horizontal lines y = c and y = d is rotated about the y-axis, the volume is
V = π ∫cd [g(y)]² dy
It is vital to express the radius in terms of y. If the original curve is given as y = f(x), you must either invert the function (on a monotonic interval) to obtain x = f−1(y), or integrate with respect to x using the shell method (not required in SL but might appear in HL). Many IB problems supply the relation in the form x = g(y) to simplify the set‑up.
当由 x = g(y)、y 轴以及水平线 y = c、y = d 围成的区域绕 y 轴旋转时，体积为
V = π ∫cd [g(y)]² dy
这里必须将半径表示为关于 y 的函数。如果原始曲线是 y = f(x) 的形式，你需要（在单调区间上）反解出 x = f−1(y)，或者利用壳层法对 x 积分（SL不考，HL可能出现）。很多 IB 题目会直接给出 x = g(y) 的形式以简化列式。
8. Volume of Revolution: Washer Method | 旋转体体积：垫圈法
Consider the region between two curves y = f(x) (outer) and y = g(x) (inner) from x = a to x = b. When this region is rotated about the x-axis, a solid with a cavity is produced. Its volume is given by the washer formula:
V = π ∫ab ( [f(x)]² − [g(x)]² ) dx
Always ensure f(x) ≥ g(x) ≥ 0 (or that the absolute radii are correctly identified). For rotations about the y-axis, the analogous formula uses x = f(y) and x = g(y): V = π ∫cd ( [f(y)]² − [g(y)]² ) dy. When the inner curve is the axis itself (g=0), the washer formula reduces to the disk method.
考虑两条曲线 y = f(x)（外）和 y = g(x)（内）在 x = a 至 x = b 之间的区域。将其绕 x 轴旋转会生成一个中空的立体，体积用垫圈公式计算：
V = π ∫ab ( [f(x)]² − [g(x)]² ) dx
务必保证 f(x) ≥ g(x) ≥ 0，或者正确识别外半径与内半径的绝对值。若绕 y 轴旋转，类似公式使用 x = f(y) 和 x = g(y)：V = π ∫cd ( [f(y)]² − [g(y)]² ) dy。当内曲线正是坐标轴（g=0）时，垫圈公式就退化为圆盘法。
9. Parametric Curves and Volumes of Revolution | 参数方程下的面积与旋转体体积
For IB AA HL students, curves defined parametrically by x = x(t), y = y(t) for t ∈ [t₁, t₂] can also be used. The area under the parametric curve (with respect to the x-axis) is A = ∫t₁t₂ y(t) x'(t) dt. If the region is rotated about the x-axis, the volume is V = π ∫t₁t₂ [y(t)]² x'(t) dt. For rotation about the y-axis, the volume is V = π ∫t₁t₂ [x(t)]² y'(t) dt. Note that the direction of integration matters: if x'(t) or
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