📚 Antidifferentiation and Area | 反微分与面积
Antidifferentiation, also known as integration, is the reverse process of differentiation. In the IB Mathematics syllabus, mastering antidifferentiation is essential for calculating areas under curves, solving differential equations, and understanding the accumulation of change. This article explores the principles of finding antiderivatives, the notation of indefinite and definite integrals, and how these concepts connect to the geometric interpretation of area.
反微分，也称积分，是微分的逆过程。在IB数学课程中，掌握反微分对于计算曲线下的面积、求解微分方程以及理解变化的累积至关重要。本文将探讨寻找反导数的原理、不定积分与定积分的符号表示，以及这些概念如何与面积的几何意义相联系。
1. What is Antidifferentiation? | 什么是反微分？
Given a function f(x), an antiderivative is any function F(x) such that F' ause the derivative of x² is 2x. Importantly, antiderivatives are not unique: adding any constant C yields another valid antiderivative, since the derivative of a constant is zero. Thus, the family of functions x² + C forms all antiderivatives of 2x.
给定函数 f(x)，其反导数就是满足 F'(x) = f(x) 的任意函数 F(x)。例如，若 f(x) = 2x，则 F(x) = x² 是一个反导数，因为 x² 的导数为 2x。重要的是，反导数并不唯一：加上任意常数 C 仍是一个有效的反导数，因为常数的导数为零。因此，函数族 x² + C 构成了 2x 的所有反导数。
2. The Indefinite Integral and the Constant of Integration | 不定积分与积分常数
d to account for the lost constant during differentiation. For instance, ∫ 2x dx = x² + C.
记号 ∫ f(x) dx 表示 f(x) 的不定积分，即其所有反导数的集合。符号 ∫ 称为积分号，f(x) 是被积函数，dx 表示积分变量。必须始终添加积分常数 +C，以弥补微分过程中丢失的常数信息。例如，∫ 2x dx = x² + C。
3. Basic Antiderivative Rules | 基本反导数法则
Memorising standard integral formulas is essential. The table below summarises the most common antiderivatives used in IB Mathematics. These rules assume the +C constant and are valid for all applicable domains.
记忆标准积分公式至关重要。下表总结了IB数学中最常用的反导数，所有公式均需加上常数 +C，并在相应定义域内有效。
| ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1 | 幂函数积分 |
| ∫ 1/x dx = ln|x| + C | 倒数积分 |
| ∫ eˣ dx = eˣ + C | 指数函数积分 |
| ∫ aˣ dx = aˣ/ln a + C | 一般指数积分 |
| ∫ sin x dx = -cos x + C | 正弦积分 |
| ∫ cos x dx = sin x + C | 余弦积分 |
| ∫ sec² x dx = tan x + C | 正割平方积分 |
m by term: ∫ [a·f(x) + b·g(x)] dx = a∫ f(x) dx + b∫ g(x) dx. This rule is used repeatedly.
此外，线性性质允许我们逐项积分和与数乘：∫ [a·f(x) + b·g(x)] dx = a∫ f(x) dx + b∫ g(x) dx。这一法则被反复使用。
4. The Definite Integral: Definition and Notation | 定积分：定义与符号
he x-axis contribute positively, while areas below contribute negatively. The definite integral can be defined as the limit of Riemann sums, but in practice we evaluate it using the Fundamental Theorem of Calculus.
定积分记为 ∫ₐᵇ f(x) dx，其中 a 和 b 是积分下限和上限。从几何上看，它表示曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的净有向面积。x 轴上方的面积贡献正值，下方的面积贡献负值。定积分可定义为黎曼和的极限，但实践中我们通过微积分基本定理来计算。
5. The Fundamental Theorem of Calculus | 微积分基本定理
x, we find an antiderivative F(x) = x². Then ∫₁³ 2x dx = [x²]₁³ = (3)² - (1)² = 9 - 1 = 8.
微积分基本定理（FTC）将反微分与面积联系起来。如果 F 是 f 在 [a,b] 上的任意反导数，则 ∫ₐᵇ f(x) dx = F(b) - F(a)。这个精妙的结果使得无需求和极限即可精确计算面积。例题：计算 ∫₁³ 2x dx，先求出反导数 F(x) = x²，则 ∫₁³ 2x dx = [x²]₁³ = 3² - 1² = 9 - 1 = 8。
and integration are inverse operations. The derivative of the integral function A(x) = ∫ₐˣ f(t) dt is f(x). This conceptual link is fundamental in both AA and AI courses.
该定理还验证了微分与积分互为逆运算。积分函数 A(x) = ∫ₐˣ f(t) dt 的导数就是 f(x)。这种概念上的联系在 AA 和 AI 课程中都至关重要。
6. Computing Areas Under a Curve (Non-negative Case) | 计算曲线下的面积（非负情形）
power rule, an antiderivative is x³/3, so area = [x³/3]₀² = (8/3) - 0 = 8/3 square units.
当在 [a,b] 上 f(x) ≥ 0 时，定积分 ∫ₐᵇ f(x) dx 直接给出由曲线、x 轴和垂直线 x = a 与 x = b 所围成的面积。例如，曲线 y = x² 下从 0 到 2 的面积是 ∫₀² x² dx。运用幂法则，一个反导数是 x³/3，因此面积 = [x³/3]₀² = 8/3 - 0 = 8/3 平方单位。
7. Area When the Curve Crosses the x-axis | 曲线跨越 x 轴时的面积
must partition the interval at the zeros of f and integrate the absolute value, i.e., ∫ₐᵇ |f(x)| dx. Consider f(x) = x³ - x on [-1,1]. Zeros occur at x = -1, 0, 1. The total area = |∫₋₁⁰ (x³ - x) dx| + |∫₀¹ (x³ - x) dx| tegral from 0 to 1 gives -1/4. Taking absolute values yields total area = 1/4 + 1/4 = 1/2.
如果 f(x) 在区间内变号，定积分给出的是净面积，而非总面积。要计算总面积，必须在 f 的零点处分段，并对绝对值积分，即 ∫ₐᵇ |f(x)| dx。考虑 f(x) = x³ - x 在 [-1,1] 上。零点是 x = -1, 0, 1。总面积 = |∫₋₁⁰ (x³ - x) dx| + |∫₀¹ (x³ - x) dx|。计算每段：∫₋₁⁰ (x³ - x) dx = [x⁴/4 - x²/2]₋₁⁰ = 0 - (1/4 - 1/2) = 1/4；第二个从 0 到 1 的积分得 -1/4。取绝对值后总面积为 1/4 + 1/4 = 1/2。
8. Area Between Two Curves | 两条曲线之间的面积
b, where f(x) ≥ g(x), is given by ∫ₐᵇ [f(x) - g(x)] dx. Always identify which function is 'upper' and which is 'lower' = ∫₋₁² [(x + 2) - x²] dx = [x²/2 + 2x - x³/3]₋₁² = (2 + 4 - 8/3) - (1/2 - 2 + 1/3) = 9/2.
曲线 y = f(x) 与 y = g(x) 在 x = a 到 x = b 之间所围成的面积，若 f(x) ≥ g(x)，则由 ∫ₐᵇ [f(x) - g(x)] dx 给出。务必先确定区间上哪条是“上曲线”，哪条是“下曲线”。如果曲线相交，通过解方程 f(x) = g(x) 找到交点以确定积分限 a 和 b。例题：求 y = x + 2 与 y = x² 之间的面积。解 x + 2 = x² 得 x = -1 和 x = 2。在 [-1,2] 上，直线位于抛物线上方，因此面积 = ∫₋₁² [(x + 2) - x²] dx = [x²/2 + 2x - x³/3]₋₁² = (2 + 4 - 8/3) - (1/2 - 2 + 1/3) = 9/2。
9. Integration by Substitution | 换元积分法
r finding antiderivatives of composite functions. The idea is to set u = g(x), then du = g' + C. For definite integrals, remember to change the limits to u-values. Example: ∫₀¹ 2x·ex² . Set u = x², du = 2x dx. When x = 0, u = 0; when x = 1, u = 1. The integral becomes ∫₀¹ eu du = [e^
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