Newton–Leibniz Formula

The upper-limit integral function is valuable because it offers a natural antiderivative candidate, linking definite integrals with differentiation—the essence of the Newton–Leibniz formula.

积分上限函数与牛顿-莱布尼茨公式

The upper-limit integral function underpins the formula:

1. Construct an antiderivative: $F(x) = \int_a^x f(t) \, dt$ is an antiderivative of $f(x)$.
2. Key fact: Any other antiderivative differs from $F(x)$ by a constant.
3. Thus: $\int_a^b f(x) \, dx = F(b) - F(a)$.

总结与理解要点

1. 几何意义

• $F(x) = \int_a^x f(t) dt$ is the accumulated area from $a$ to $x$.
• As $x$ increases, the accumulated area grows.
• $F(x)$ equals the area under $f(x)$ from $a$ to $x$.

2. 微分关系

• Core property: $F'(x) = f(x)$.
• The derivative of the upper-limit integral equals the integrand.
• This builds the bridge between integration and differentiation.

3. 原函数关系

• The upper-limit integral is an antiderivative of the integrand.
• Any antiderivative differs from it by a constant.
• This is the key premise of the Newton–Leibniz formula.

4. 应用价值

• Provides a route to compute definite integrals.
• Connects to differential equations and cumulative quantities in physics.
• Forms the theoretical basis of the Fundamental Theorem of Calculus.

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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）	积分中值定理中位于区间 $[x,x+h]$ 的取值点
$\int$	数学符号	Integral（积分符号）	表示累积面积或累积量的定积分运算
$F(x)$	函数符号	Function（函数）	积分上限函数，表示从下限到 $x$ 的累积面积

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/	以 $x$ 作为积分上限、描述累积面积的函数
变限积分	variable-limit integral	/ˈvɛəriəbəl ˈlɪmɪt ˈɪntɪɡrəl/	上下限随 $x$ 变化的积分形式
积分中值定理	mean value theorem for integrals	/miːn ˈvæljuː ˈθɪərəm fɔːr ˈɪntɪɡrəlz/	存在 $\xi$ 使积分等于函数值乘区间长度
累积量	accumulated quantity	/əˈkjuːmjəˌleɪtɪd ˈkwɒntɪti/	由瞬时变化率积分得到的总量

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