📚 Areas between curves and lines | 曲线与直线间的面积
Integration is a powerful tool for calculating the area of regions bounded by curves and straight lines. In A-Level Mathematics, you must be able to set up and evaluate definite integrals to find areas enclosed by one or more functions, carefully handling intersections and sign changes to avoid negative results. This article walks you through the key concepts, from basic area under a curve to handling regions between two curves, using clear examples and helpful strategies.
积分是计算由曲线与直线所围成区域面积的有力工具。在A-Level数学中，你需要能够建立并计算定积分，求出由一个或多个函数围成的面积，并谨慎处理交点与符号变化，以避免得到负值。本文将从基本的曲线下方面积开始，逐步延伸到处理两条曲线之间的区域面积，配以清晰的例子和实用的策略。
1. Introduction to Finding Areas by Integration | 积分法求面积简介
(x) and the x-axis over an interval [a, b]. The fundamental idea is that the area is given by ∫ₐᵇ f(x) dx, provided f(x) does not change sign. If the curve dips below the x-axis, the integral yields a negative value, which must be handled by splitting the interval or taking absolute values.
定积分可以计算在区间[a, b]上曲线 y = f(x) 与 x 轴之间的净面积。基本思想是，若 f(x) 不变号，面积由 ∫ₐᵇ f(x) dx 给出。如果曲线落在 x 轴下方，积分将给出负值，此时必须通过分割区间或取绝对值来处理。
For more complex shapes bounded by several curves, we generalise this idea by integrating the difference between the upper and lower boundaries with respect to x (or right minus left with respect to y). Visualising the region and identifying the correct limits are the first essential steps.
对于由多条曲线围成的更复杂图形，我们将这一思想推广为对 x 积分上方函数减去下方函数（或对 y 积分右方减左方）。可视化区域并确定正确的积分限是最关键的初始步骤。
2. Area Between a Curve and the x-Axis | 曲线与x轴之间的面积
When y = f(x) is entirely above the x-axis between x = a and x = b, the area is simply A = ∫ₐᵇ f(x) dx nd f(x) ≤ 0. The total area is the sum of the absolute values of these separate integrals.
当 y = f(x) 在 x = a 与 x = b 之间完全位于 x 轴上方时，面积即为 A = ∫ₐᵇ f(x) dx。如果曲线穿过 x 轴，则需要求出 f(x) = 0 的点，并将积分分割为 f(x) ≥ 0 与 f(x) ≤ 0 的区间。总面积等于各部分积分绝对值之和。
For instance, to find the area bounded by y = x² − 4 and the x‑ rst and adding it to the second. Forgetting to take the absolute value is one of the most common mistakes.
例如，计算 y = x² − 4 与 x 轴在 x = 0 到 x = 3 之间所围成的面积时，首先注意到曲线在 x = 2 处穿过 x 轴。然后分别计算 ∫₀² (x² − 4) dx 和 ∫₂³ (x² − 4) dx，取第一个积分的绝对值并与第二个相加。忘记取绝对值是最常见的错误之一。
3. Area Between a Curve and the y-Axis | 曲线与y轴之间的面积
x = g(y) and lies entirely to the right of the y-axis between y = c and y = d, the area is A = ∫꜀ᵈ g(y) dy. When the curve crosses the y-axis, split the integral at the crossing points and take absolute values, just as with x‑axis areas.
有时对 y 积分更为方便。若曲线表示为 x = g(y) 并且在 y = c 到 y = d 之间完全位于 y 轴右侧，面积即为 A = ∫꜀ᵈ g(y) dy。当曲线穿过 y 轴时，同样需要在交点处分割积分并取绝对值，与 x 轴面积的处理方式一致。
ndaries include horizontal lines. For example, the area between the curve x = y² and the y‑axis from y = 0 to y = 2 is ∫₀² y² dy = [y³/3]₀² = 8/3.
当函数比较容易表示为 x 关于 y 的表达式，或者边界包含水平直线时，这种方法尤为方便。例如，曲线 x = y² 与 y 轴在 y = 0 到 y = 2 之间的面积为 ∫₀² y² dy = [y³/3]₀² = 8/3。
4. Area Between a Curve and a Line | 曲线与直线之间的面积
ea is generally found by integrating the difference between the upper and lower functions over the x‑ their intersection points. Always sketch the region to identify which function is on top.
当区域由曲线 y = f(x) 与直线 y = mx + c 围成时，面积通常通过在其交点确定的 x 区间上，对上方函数与下方函数的差进行积分来求得。务必先绘制区域草图，以明确哪个函数在上方。
The area formula is A = ∫ₐᵇ [top − bottom] dx, where a and b are the x‑coordinates of the points of intersection. If the line is above the curve in one part and below in another, you need to split the integral at the point where they cross.
面积公式为 A = ∫ₐᵇ [上方 − 下方] dx，其中 a 与 b 为交点的横坐标。如果直线在部分区间位于曲线上方，而在另一部分位于下方，则需在它们相交的点处分割积分。
, 2. To find the total enclosed area, examine the position on [−2,0] and [0,2] separately.
假设曲线为 y = x³ − 3x，直线为 y = x。解方程 x³ − 3x = x 得到 x³ − 4x = 0 → x(x−2)(x+2)=0，因此交点为 x = −2, 0, 2。要计算总围成面积，需分别考察区间 [−2,0] 和 [0,2] 上的上下关系。
5. Area Between Two Curves | 两条曲线之间的面积
ves y = f(x) and y = g(x) from x = a to x = b, where f(x) ≥ g(x) on the whole interval, is A = ∫ₐᵇ [f(x) − g(x)] dx. The curves can be any functions, not necessarily a straight line and a curve.
在区间 [a, b] 上，若总有 f(x) ≥ g(x)，两条曲线 y = f(x) 与 y = g(x) 之间的面积公式为 A = ∫ₐᵇ [f(x) − g(x)] dx。这里的曲线可以是任意函数，不一定非要一条是直线。
When f and g swap vertically, the interval must be broken at the x‑ table for the difference h(x) = f(x) − g(x) will reveal where the upper function changes.
当 f 与 g 的上下关系发生交换时，必须在方程 f(x) = g(x) 的解对应的 x 值处分割区间。通过快速绘制草图或列出差函数 h(x) = f(x) − g(x) 的符号表，可以明确上方函数是在何处发生改变的。
or curves expressed as x = f(y) and x = g(y), the area between them from y = c to y = d is A = ∫꜀ᵈ [right − left] dy, provided one function is always to the right of the other on the interval.
对于形如 x = f(y) 和 x = g(y) 的曲线，如果在区间 [c, d] 上某一函数始终位于另一函数的右侧，则它们之间的面积公式为 A = ∫꜀ᵈ [右方 − 左方] dy。
6. Determining Integration Limits by Solving Equations | 通过解方程组确定积分限
Finding the limits of integration is a critical step. For regions bounded by curves and lines, set the equations equal and solve for the variable you plan to integrate with respect to. This gives the intersection points.
确定积分限是关键步骤。对于由曲线和直线围成的区域，令方程相等并解出你想对其进行积分的变量，即可得到交点。
which function is greater in between by testing a sample point, say x = 0: 2(0)² = 0, 4 − 0 = 4, so the line is above.
例如，求由 y = 2x² 与 y = 4 − 2x 所围成区域的面积，可解方程 2x² = 4 − 2x → 2x² + 2x − 4 = 0 → x² + x − 2 = 0 → (x+2)(x−1)=0，得到积分限 x = −2 与 x = 1。取区间内一点检验大小关系，例如 x = 0：2(0)² = 0，4 − 0 = 4，因此直线在上方。
If the question asks for the area completely enclosed by curves, there may be more than two intersection points. Always solve for all intersections and order them to define the correct intervals.
如果题目要求计算由曲线完全围成的封闭区域面积，可能有多于两个交点。务必解出所有交点，并按顺序排列以确定正确的积分区间。
7. Handling Curve Intersections and Splitting Regions | 处理曲线交叉与区域分割
When two curves intersect more than twice or when the 'top' function changes, you cannot apply a single integral over the whole range. Divide the total area into sub‑regions at each intersection point and compute the area of each sub‑region separately, then sum the results.
当两条曲线相交超过两次，或者“上方”函数发生变化时，就不能对整个范围只用一个积分。应在每个交点处将总面积分割成若干子区域，分别计算各个子区域的面积，再求和。
al enclosed area is ∫₋₁⁰ (x³ − x) dx + ∫₀¹ (x − x³) dx. This splitting avoids a negative contribution from a straightforward ∫₋₁¹ (x³ − x) dx, which would give zero.
考虑 y = x³ 与 y = x，它们相交于 x = −1, 0, 1。在 [−1, 0] 上，x³ 位于 x 上方（因为 (−0.5)³ = −0.125 > −0.5），而在 [0, 1] 上，x 位于 x³ 上方。总面积为 ∫₋₁⁰ (x³ − x) dx + ∫₀¹ (x − x³) dx。这样分割可以避免只计算 ∫₋₁¹ (x³ − x) dx 得到零而忽略了实际面积。
Always sketch the graphs to see the crossing points and the relative positions. Even a rough sketch will prevent sign errors.
一定要画出函数的草图，观察交点及其相对位置。即使是简图也有助于避免符号错误。
8. Simplifying with Symmetry and Absolute Values | 利用对称性和绝对值简化计算
If a region is symmetric about the y‑ ute the area on one side and double it. For example, the area between y = 4 − x² and the x‑axis is symmetric; you can integrate from 0 to 2 and multiply by 2.
如果区域关于 y 轴对称，你可以只计算一侧的面积再乘以2。例如，y = 4 − x² 与 x 轴之间的面积即是对称的；可以从 0 积分到 2，然后将结果乘以2。
A more systematic approach for avoiding sign errors is to integrate the absolute difference: Area = ∫ₐᵇ |f(x) − g(x)| dx. In many cases, you can remove the absolute value by identifying the intervals where f(x) − g(x) is positive or negative, but writing the integral with absolute values reminds you to account for sign changes.
一种更能避免符号错误的方法是对绝对差值进行积分：Area = ∫ₐᵇ |f(x) − g(x)| dx。虽然可以通过确定 f(x) − g(x) 的正负区间来去掉绝对值，但用绝对值形式书写积分可以提醒你考虑符号变化。
Some calculators and software can handle |f(x) − g(x)| directly, but in an exam setting you must still show the splitting work. Use the absolute value notation as a thinking tool.
某些计算器和软件可以直接处理 |f(x) − g(x)|，但在考试中仍须展示分割过程。可将绝对值符号作为思维工具使用。
9. Worked Examples and Common Pitfalls | 典型例题与常错点
Example: Find the area enclosed by the curve y = x² and the line y = x + 2.
例题： 求曲线 y = x² 与直线 y = x + 2 所围成区域的面积。
ay x = 0: line gives 2, curve gives 0, so line is above. The area is ∫₋₁² [(x+2) − x²] dx.
首先求交点：x² = x + 2 → x² − x − 2 = 0 → (x−2)(x+1)=0，得 x = −1, 2。取中间一点测试，如 x = 0：直线值为2，曲线值为0，故直线在上方。面积为 ∫₋₁² [(x+2) − x²] dx。
1/3 = −7/6. Difference: 10/3 − (−7/6) = 20/6 + 7/6 = 27/6 = 9/2. Area is 4.5 square units.
计算：∫ (x+2−x²) dx = [x²/2 + 2x − x³/3] 从 −1 到 2。在 x=2：4/2+4−8/3=2+4−8/3=10/3。在 x=−1：1/2−2+1/3=−7/6。相减：10/3 − (−7/6) = 20/6+7/6=27/6=9/2。面积为4.5平方单位。
Common pitfalls:
常见错误：
Forgetting to subtract the lower function: just integrating the upper curve gives the wrong answer.
忘记减去下方函数：仅对上方曲线积分将得到错误答案。
Using the wrong limits: always solve for intersections; do not assume limits are the given endpoints unless the region is explicitly outlined.
积分限错误：务必解出交点；除非题目明确给出边界，否则不要假设积分限就是给定的端点。
Not splitting the integral when the curves cross: this causes cancellation of areas and yields a net area rather than a total area.
曲线相交时未分割积分：这会导致面积正负抵消，得到的是净面积而非总面积。
Misidentifying which function is on top: always test a number between intersections.
上下关系判断错误：一定要在交点之间取一个数值进行检验。
For area with respect to y, confusing right and left: sketch and use right − left.
对 y 积分时混淆左右：应画草图并使用 右方 − 左方。
Practice with a variety of curve–curve and curve–line combinations, and always draw a quick graph, even if only a rough sketch, to visualise the region and the upper/lower functions.
多练习各种曲线与曲线、曲线与直线的组合题目，并始终快速画出示意图（哪怕是简图），以便直观地看到区域及上下函数关系。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图