Antidifferentiation and Area Under a Curve: Introductory Concepts | 反微分与曲线下面积：初步概念

📚 Antidifferentiation and Area Under a Curve: Introductory Concepts | 反微分与曲线下面积：初步概念

In differential calculus, differentiation tells us how a function changes instantaneously. Integral calculus answers the reverse question: if we know the rate of change, can we recover the original function? This process is called antidifferentiation. Surprisingly, the same idea gives us a way to measure the area under a curve.

在微分学中，微分告诉我们函数如何瞬时变化。积分学则要回答相反的问题：如果已知变化率，我们能否还原出原来的函数？这个过程称为反微分。令人惊讶的是，同一个想法还为我们提供了测量曲线下面积的方法。

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1. What is Antidifferentiation? | 什么是反微分？

If a function F has derivative f, that is F'(x) = f(x), then F is called an antiderivative of f. For example, because the derivative of x² is 2x, x² is an antiderivative of 2x.

若函数 F 的导数为 f，即 F'(x) = f(x)，则称 F 是 f 的一个反导数（原函数）。例如，由于 x² 的导数是 2x，所以 x² 是 2x 的一个反导数。

Antidifferentiation is the inverse operation of differentiation. It is also called integration, and the process produces the indefinite integral.

反微分是微分的逆运算。它也被称为积分，其过程产生不定积分。

F'(x) = f(x) ⇔ ∫ f(x) dx = F(x) + C

Here C is the constant of integration, because differentiating any constant gives 0.

这里 C 是积分常数，因为任何常数求导后都得到 0。

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2. The Indefinite Integral and Constant of Integration | 不定积分与积分常数

The symbol ∫ f(x) dx denotes the indefinite integral of f. It represents the family of all antiderivatives of f. Because the derivative of a constant is zero, every indefinite integral must include “+ C”.

符号 ∫ f(x) dx 表示 f 的不定积分。它代表 f 的所有反导数组成的族。由于常数的导数为零，每个不定积分都必须包含 “+ C”。

For instance, ∫ 2x dx = x² + C. The constant C allows for infinitely many vertical shifts of the curve, all with the same slope at every x.

例如，∫ 2x dx = x² + C。常数 C 允许曲线在垂直方向上有无穷多种平移，但在每个 x 处的斜率都相同。

The derivative of x², x² + 5, and x² − 7 are all 2x, which is why the family is written with C.

x²、x² + 5 和 x² − 7 的导数都是 2x，因此这个函数族用 C 表示。

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3. Basic Rules of Antidifferentiation | 反微分的基本法则

Just as differentiation has rules, antidifferentiation has corresponding rules. Below are the most essential ones.

正如同微分有运算规则一样，反微分也有相应的法则。以下是若干最基本的规则。

f(x)	∫ f(x) dx
xⁿ (n ≠ −1)	xⁿ⁺¹/(n+1) + C
1/x	ln|x| + C
eˣ	eˣ + C
sin x	−cos x + C
cos x	sin x + C

The power rule states: if n ≠ −1, then ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C. For n = −1, the antiderivative is ln|x| + C.

幂法则指出：若 n ≠ −1，则 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。当 n = −1 时，其反导数为 ln|x| + C。

Antidifferentiation is linear: ∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx.

反微分具有线性性质：∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx。

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4. From Antiderivatives to Areas | 从反导数到面积

How does antidifferentiation relate to area? Consider a continuous function f that is positive on [a, b]. To approximate the area under the curve, we can split [a, b] into thin rectangles of width Δx and height f(xᵢ), where xᵢ is a sample point in each subinterval.

反微分与面积有什么关系？考虑一个在 [a, b] 上非负的连续函数 f。为了近似曲线下的面积，我们可以把 [a, b] 分成许多宽度为 Δx、高度为 f(xᵢ) 的细长矩形，其中 xᵢ 是每个子区间中的取样点。

The total approximate area is the sum ∑ f(xᵢ) Δx. As Δx → 0, this sum approaches the exact area, which is written as the definite integral ∫ₐᵇ f(x) dx.

总面积近似值为 ∑ f(xᵢ) Δx。当 Δx → 0 时，这个和趋近于精确面积，记为定积分 ∫ₐᵇ f(x) dx。

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5. The Definite Integral: Signed Area | 定积分：有符号面积

The notation ∫ₐᵇ f(x) dx is called a definite integral. The numbers a and b are the lower and upper limits of integration, and f(x) is the integrand. Unlike the indefinite integral, it produces a number (when it exists).

记号 ∫ₐᵇ f(x) dx 称为定积分。a 和 b 分别称为积分下限和上限，f(x) 称为被积函数。与不定积分不同，定积分（若存在）得到一个数值。

This number represents the “signed area” between the graph of f and

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