Area View

The upper-limit integral function is an entry point to definite integrals. The best way to master it is to see the “area accumulation” intuition, then watch how the function changes as the upper limit moves.

从面积变化理解积分上限函数

场景：水库蓄水问题

Imagine a reservoir with inflow at a varying rate. Let $f(t)$ be the inflow rate at time $t$ (m³/hour). The total stored water from $0$ to $x$ is:

思考过程：

1. Instantaneous inflow at time $t$ is $f(t)$。
2. Over a tiny interval $dt$, inflow volume is approximately $f(t)\cdot dt$。
3. Total inflow from $0$ to $x$ is $\int_0^x f(t)\,dt$, which depends on $x$:

$F(x) = \int_0^x f(t)\,dt$

This is the upper-limit integral function: the accumulated area (or quantity) from the start up to $x$.

基础可视化：面积累积过程

高级可视化：积分上限函数的完整理解

Observations from the interactive chart:

1. Blue curve: original function $f(x) = \sin(x) + 1$ — the instantaneous rate.
2. Green curve: upper-limit integral $F(x) = \int_0^x f(t)\,dt$ — the accumulated quantity.
3. Blue shaded area: integral area from $0$ to $x$.
4. Red dotted line: derivative $F'(x)$, coinciding with $f(x)$.

Key insight: the derivative of the upper-limit integral equals the integrand, i.e., $F'(x) = f(x)$.

更多生活化例子

例子 1：汽车行驶距离

• $f(t)$: instantaneous speed at time $t$
• $F(x) = \int_0^x f(t)\,dt$: total distance from start to time $x$
• $F'(x) = f(x)$: derivative of distance is speed

例子 2：人口增长

• $f(t)$: population growth rate at time $t$
• $F(x) = \int_0^x f(t)\,dt$: total population increase up to time $x$
• $F'(x) = f(x)$: derivative of total increase is the growth rate

例子 3：经济收入

• $f(t)$: income rate at time $t$
• $F(x) = \int_0^x f(t)\,dt$: total income up to time $x$
• $F'(x) = f(x)$: derivative of total income is the income rate

All illustrate the same principle: the derivative of an accumulated quantity equals the instantaneous rate.

总结

本文出现的符号

中英对照