Definition and Derivative Property

After the area intuition, we formalize the upper-limit integral function and prove its key property: its derivative equals the integrand.

定义

Upper-limit integral function

Let $f(x)$ be continuous on $[a, b]$. Define

$F(x) = \int_a^x f(t) \, dt$

This is the upper-limit integral function.

基本性质

定理 1

For $F(x) = \int_a^x f(t) \, dt$ on $[a, b]$, $F$ is differentiable and

$F'(x) = f(x)$

This is the “first part” of the Fundamental Theorem of Calculus, linking differentiation and integration.

证明的详细过程

步骤 1：利用导数的定义

$F'(x) = \lim_{h \to 0} \frac{F(x+h) - F(x)}{h} = \lim_{h \to 0} \frac{\int_a^{x+h} f(t) \, dt - \int_a^x f(t) \, dt}{h}$

步骤 2：利用积分的区间可加性

$F'(x) = \lim_{h \to 0} \frac{\int_x^{x+h} f(t) \, dt}{h}$

步骤 3：利用积分中值定理

There exists $\xi \in [x, x+h]$ (or $[x+h, x]$) such that

$\int_x^{x+h} f(t) \, dt = f(\xi) \cdot h$

步骤 4：取极限

$F'(x) = \lim_{h \to 0} \frac{f(\xi) \cdot h}{h} = \lim_{h \to 0} f(\xi) = f(x)$

The proof is complete. The upper-limit integral function has clear geometric meaning and is differentiable, enabling the applications that follow.

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\xi$	希腊字母	Xi（ksee）	积分中值定理中的区间内取值点
$F'(x)$	数学符号	导数	积分上限函数的导数，等于被积函数

中英对照

中文术语	英文术语	音标	说明
积分上限函数	upper-limit integral function	/ˈʌpər ˈlɪmɪt ɪnˈtɛɡrəl ˈfʌŋkʃən/	形如 $F(x)=\int_a^x f(t)\,dt$ 的函数
积分中值定理	mean value theorem for integrals	/miːn ˈvæljuː ˈθɪərəm fɔːr ˈɪntɪɡrəlz/	保证存在 $\xi$ 使积分等于 $f(\xi)$ 乘区间长度
微积分基本定理	Fundamental Theorem of Calculus	/ˌfʌndəˈmɛntl ˈθɪərəm əv ˈkælkjʊləs/	连接导数与定积分的核心定理

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