Integrals

Core Sentence / 核心句

An integral accumulates many tiny pieces into a total amount.

中文理解：积分的核心不是符号，而是把很多小量累加成总量。

Formula / 公式或规则

$${\int }_a^{b}f(x)dx$$

In area language, this adds thin rectangle-like pieces under a curve. In physics, it often adds tiny contributions such as $dE$, $dq$, or $dW$.

When To Use / 什么时候用

看到 total、accumulated amount、area、continuous sum、很多微小贡献相加，就想到 integral。

Watch Out / 易错点

不要只把积分理解成求面积；面积只是 accumulation 的一种可视化。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• Chapter 9 · Continuous Average
  • integral converts dense finite sampling into signed area divided by interval width
• Chapter 8 · Integral as a Limit
  • continuous accumulation is the limit of increasingly fine Riemann sums
• AP电磁学第1次正课-电荷库仑定律与电场 · Derivatives and Integrals / 微积分工具
  • Derivatives and Integrals / 微积分工具
• 01 - The Essence of Calculus · Knowledge Point 6: Derivative and Integral Undo Each Other
  • Derivatives and integrals are inverses of each other.
• 03 - Derivative Formulas through Geometry · 1. What This Episode Is About / 这一集在讲什么
  • Chapter 1 先讲 Integrals 和 accumulated area；Chapter 2 解释 Derivatives 到底是什么意思：不是神秘的 instantaneous change，而是 Limits 下的 Average Rate of Change。
• 02 - The Paradox of the Derivative · 2. Link to Previous Episode / 和上一集的连接
  • Chapter 1: why integrals and derivatives are connected
• AP电磁学第1次正课-电荷库仑定律与电场 · 2. Class Flow / 课堂脉络
  • 1. 课前工具：Vectors、derivatives/integrals、kinematics、Newton's Second Law、energy and momentum。
• 微积分BC第7次正课 · 积分为何出现
  • finite local contributions become exact continuous accumulation through a limit
• 微积分BC第8次正课 · FTC
  • antiderivative endpoint change computes exact accumulation
• 微积分BC第9次正课 · 方法选择
  • algebraic structure determines which antiderivative technique simplifies the integrand

Connected Notes / 连接

• 01 - The Essence of Calculus
• 08 - Integration and the Fundamental Theorem of Calculus
• 09 - What Does Area Have to Do With Slope
• Average Value of a Function
• Riemann Sum
• Antiderivative
• Signed Area
• Area Under a Curve
• Fundamental Theorem of Calculus
• Integral Test
• AP电磁学第2次正课-连续带电体电场与电通量
• Definite Integral
• 微积分BC第7次正课-导数图像优化与定积分入门
• Accumulation Function
• U-Substitution
• 微积分BC第8次正课-反导FTC变上限积分与换元
• Integration by Parts
• Partial Fractions
• Improper Integral
• 微积分BC第9次正课-积分方法与反常积分