微积分（积分到微分方法） · free preview
The Integral of Step Functions
回到阶梯函数面积公式 A(s)=ΣcₖΔxₖ，用适配分割重新解读每一项；通过加细实验与动手计算亲眼见证「刀在变，和不变」，亲手补全「插入一刀、总和不变」的证明，得出分割不变性；进而删除分割信息，只保留从哪开始、到哪结束、累加什么，给出阶梯函数积分 ∫ₐᵇ s(x)dx 的严格定义，理解 ∫ 是拉长的 S、dx 是记录宽度的占位符。
1. Step 1 · The knife changes, the sum does not: the integral of step functions
Step 1 · The knife changes, the sum does not: the integral of step functions
2. Interactive tool: th…
Interactive tool: the refinement experiment. Keep clicking "Refine once" to add a random extra cut inside some open subinterval, and watch how the expansion of A(s) above and the computed result change.
3. Hands-on practice: c…
Hands-on practice: compute the area sum beneath the step function in the figure using the refined partition P₂ = {0, 2, 3.5, 5}. (Recall: the coarse partition P₁ = {0, 2, 5} gives 3·(2−0) + 1·(5−2) = 6 + 3 = 9.) What total do you get with P₂?
4. Proof walkthrough: i…
Proof walkthrough: insert a cut t into the open subinterval (xₖ₋₁, xₖ); only the k-th term is disturbed. Complete the derivation below and prove "the knife changes, the sum does not" with your own hands.
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