Area Under a Curve

Core Sentence / 核心句

Area under a curve is the visual model for accumulation.

中文理解：曲线下面积是理解积分最直观的模型：很多小面积加成总面积。

Formula / 公式或规则

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

When To Use / 什么时候用

看到 accumulated area、integral intuition、Riemann sum，就回到 area under a curve。

Watch Out / 易错点

面积图像是入口，不是积分的全部含义。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• Chapter 9 · Area Divided by Width
  • signed area divided by interval width gives average height
• Chapter 8 · Area as Accumulation Language
  • area visualizes the limit of many tiny local contributions
• 第7课 · 从面积到累积
  • irregular area motivates local approximation, but accumulation is the broader idea

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
• Signed Area
• Integrals
• Fundamental Theorem of Calculus
• Polar Area
• Definite Integral
• 微积分BC第7次正课-导数图像优化与定积分入门