Once upon a time, there was a mountain, in the mountain there was a temple, in the temple there was a tall tree, and on it hung many, many people… 【Dense Fog】
In the early stages of mathematical analysis (laughs), we learn the concepts of differentiation and integration and get a vague feeling: differentiation and integration are inverse operations of each other. Let’s re-examine this concept using a one-dimensional lens.
1. What is Differentiation?
If a function $f$ is differentiable at point $x$, we can approximate it linearly. Geometrically, we are looking at the slope.
\[f(y) = f(x) + a(y - x) + o(y - x)\]The term $o(y-x)$ is the “magic” part—it’s an error that disappears faster than the linear part as we get closer. We usually call $a$ the derivative, denoted as $f’$.
2. What is Integration?
Think of (Riemann) integration like eating a hamburger. One bite at a time, you finish the whole thing. We divide the area into tiny rectangular “bites”:
\[\lim_{\max |x_{i+1} - x_i| \to 0} \sum_{i=0}^{n-1} a(\xi_i)(x_{i+1} - x_i) = \int_0^1 a(x) \, dx\]Try it yourself! 🍔
Move the slider to see how “taking more bites” (increasing $n$) makes our approximation perfect.
3. Are They Really Inverses?
Case A: Integrate, then Differentiate
If we define $f(x) = \int_0^x a(t) \, dt$, then $f’(x)$ is just the growth rate of that area. Since the area grows by exactly $a(x)$ at that moment, we get our original function back. Success!
Case B: Differentiate, then Integrate
Using the Lagrange Mean Value Theorem, we can show that: \(f(x) - f(0) = \sum f'(\xi_i)(x_{i+1} - x_i) \to \int_0^x f'(t) \, dt\) This is the famous Newton-Leibniz formula. I just re-proved it. 2333333!
4. Wait… Is there a trap? ๑´ڡ`๑
Is it always this perfect? Consider a “step” function $a(x)$.
\[a(x) = \begin{cases} 1 & 0 \le x < 1 \\ 0 & \text{otherwise} \end{cases}\]When we integrate this to get $f(x)$, we find that $f$ has “sharp corners” at $x=0$ and $x=1$. At those corners, $f$ is not differentiable. > The Mathematician’s Dilemma: > If we have a problem at one point, do we ignore it? What about two points? Or a countable number of points? If we change a function at too many points, it might stop being Riemann integrable entirely!
