Fundamental Ideas of Integral Calculus

问题

Imagine you are a 17th-century mathematician facing a simple-looking question: How do we find the area of an irregular shape?

Examples:

Known formulas then:

But how to handle irregular shapes?

Mathematicians found a key observation:

If you slice an irregular region into many thin rectangles, then:

Thus, complex areas can be computed using simple rectangles!

矩形条数量：16

调整矩形条的数量，观察如何将不规则图形分割成矩形条来计算面积。

当矩形条越来越窄时，近似面积越来越接近真实面积。

Split $[a,b]$ into $n$ equal parts:

$[a,b]$ (closed interval): all real $x$ with $a\le x \le b$ .
$\Delta x$ (Delta x): change/width of each small interval.

Rectangle approximation

$S_n = \sum_{i=1}^{n} f(x_i)\, \Delta x$

$\sum$ (Sigma): summation, $\sum_{i=1}^{n}$ means sum from $i=1$ to $n$ .

When $n\to\infty$ (so $\Delta x \to 0$ ),

Core idea of integrals

$\text{Exact area} = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\, \Delta x$

$\lim$ (limit): as $n\to\infty$ , expressions approach a value, e.g. $\lim_{n\to\infty}\tfrac1n=0$ .

More rectangles → better approximation
Infinitely many infinitesimal rectangles → exact area
Integrals approximate complex areas by summing infinitely many tiny rectangles

矩形数量：1

当矩形数量越来越多时，近似面积越来越接近真实面积

这就是积分的几何意义：用无穷多个无穷小的矩形来精确计算面积

当前近似面积：0.0000

真实积分值：2.6667