Accumulation Function

Core Sentence / 核心句

An accumulation function assigns each endpoint the signed total accumulated from a fixed starting point to that endpoint.

中文理解：固定起点、移动终点后，每个终点都会对应一个累计量，因此积分本身成为一个函数。

Definition / 定义

$$A(x)={\int }_a^{x}f(t)dt.$$

$t$ is a dummy integration variable; $x$ is the external input controlling the endpoint.

Fundamental Rule / 核心规则

If $f$ is continuous,

$$A^{\prime }(x)=f(x).$$

For a composite upper bound,

$$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x).$$

Trigger Sentence / 触发句

The derivative of the accumulated total is the current local contribution.

累计总量的导数，就是当前位置的局部贡献。

Watch Out / 易错点

• $A(x)$ is a function, not a fixed area number.
• $A(a)=0$ even if $f(a)\ne 0$.
• $A$ increases where $f>0$, not necessarily where the graph of $f$ is rising.
• composite upper bounds require the Chain Rule factor $g^{\prime }(x)$.
• signed accumulation may decrease when $f<0$.

Source / 出现位置

• 第8课 · 变上限积分函数
• 第8课 · FTC Part 1
• Chapter 8 · Current height

Connected Notes / 连接

• Fundamental Theorem of Calculus
• Definite Integral
• Antiderivative
• Signed Area
• Chain Rule
• 微积分BC第8次正课-反导FTC变上限积分与换元
• 08 - Integration and the Fundamental Theorem of Calculus