Integration: Differentiation's Mirror Twin

The Other Half

If differentiation pulls slopes out of curves, integration pushes accumulated change back into a quantity. Differentiation answers "how fast?"; integration answers "how much, over time?"

The Fundamental Theorem of Calculus says these two are inverse operations. Integrate a derivative and you get the original function back (up to a constant). They unwind each other.

The Area-Under-the-Curve Picture

The integral $\int_{a}^{b} f (x) d x$ equals the area under the curve $f (x)$ from $x = a$ to $x = b$ . Why? Because integration is "accumulating tiny rectangles of width $d x$ and height $f (x)$ and summing them up." Take the limit as $d x \to 0$ and you get exact area.

Why AI Cares Less About Integration

Most ML loss functions are sums (already discrete) or already-averaged. We rarely need to compute symbolic integrals. Where integration shows up:

Probability distributions — a continuous probability density integrates to 1.
Variational inference — approximating intractable integrals over latent variables.
Continuous-time models — neural ODEs, diffusion models. These do involve real integration.
Reward accumulation in reinforcement learning — discounted sum of future rewards, the discrete version of an integral.

Differentiation tells you the rate; integration tells you the total. They're inverses. AI uses the first constantly, the second occasionally.

Code

Riemann sum approximation·python

import numpy as np

# Numerical integration: area under f(x) = x^2 from 0 to 1
# True answer: 1/3

def f(x): return x ** 2

# Approximate with rectangles (Riemann sum)
n = 1000
x = np.linspace(0, 1, n)
dx = 1.0 / n
area = np.sum(f(x) * dx)
print(area)         # ~0.333  (≈ 1/3)

Integration: Differentiation's Mirror Twin

The Other Half

The Area-Under-the-Curve Picture

Why AI Cares Less About Integration

Code

External links

Exercise

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