10. Integration

From the Riemann integral to the Lebesgue integral — how the idea of integration expands.

• The Gamma Function — The factorial stretched to real arguments — $\Gamma(x+1) = x\Gamma(x)$, so $\Gamma(n+1) = n!$. Watch the improper integral converge for $x > 0$.
• The Beta Function — The area under \t^{p-1}(1-t)^{q-1}\ is the beta function — morph its shape with the \p\, \q\ sliders. Also covers \B(p,q) = \\Gamma(p)\\Gamma(q)/\\Gamma(p+q)\, the bridge to the gamma function.
• Bohr–Mollerup Theorem — $f(1) = 1$ and $f(x+1) = x\,f(x)$ do not pin down the continuous extension of the factorial — the impostor $\Gamma(x)(1 + a\sin 2\pi x)$ matches $\Gamma$ at every integer and survives. Watch the secant test expose log-convexity as $\Gamma$'s true fingerprint.
• Improper Integrals — $\int_1^\infty \frac{dx}{x^p}$ converges only for $p > 1$, plus the singular kind $\int_0^1 \frac{dx}{\sqrt{x}} = 2$ — the twin of the p-series scene.
• Upper and Lower Sums Close In — Fill rectangles under $f(x) = x^2$. The lower and upper sums squeeze $\frac{1}{3}$ from both sides.
• Volterra's Function — Differentiable everywhere with a bounded derivative — yet the derivative is not Riemann integrable. A sum of $(x-a)^2(b-x)^2\sin\frac{1}{x-a}$-type bumps over a fat Cantor set builds this legendary counterexample.
• The Area Function's Derivative Is the Function — Measure the slope of the area function $A(x) = \int_a^x f(t)\,dt$ — it equals $f(x)$. Drag $x$.
• Functions of Bounded Variation — A function has bounded variation when $\sum \lvert f(x_i) - f(x_{i-1})\rvert$ stays under a finite ceiling as partitions refine — every monotone function qualifies, while $x\sin(1/x)$, wiggling endlessly near 0, does not.
• Jordan Decomposition — Every BV function is a difference of two increasing functions — $f = g - h$ with $g(x) = V_0^x(f)$ the variation function. Drag the probe point over $x^2\sin\frac{1}{x}$ and watch the identity hold pointwise.
• Lebesgue Measure — Outer measure is the cheapest countable interval cover of a set. A single point — and even all the rationals — cost nothing. Measurable sets and countable additivity need the Carathéodory construction (idea level only).
• Covering the Rationals with Nothing — Cover the rationals one by one with $\varepsilon/2^k$ covers — the total length drops below $\varepsilon$. Riemann can't move.
• Absolute Continuity — $F(x) = x^2$ is absolutely continuous — when the intervals' total length goes to 0, so does $\sum |\Delta F|$. The Cantor function is uniformly continuous but not absolutely continuous: total length $\left(\frac23\right)^n \to 0$ while $\sum |\Delta c| = 1$ stays frozen. Absolutely continuous functions are exactly the indefinite integrals of Lebesgue-integrable functions.