Cheatsheet: Integration and the fundamental theorem

The definite integral

∫_a^b f(x) dx = lim (n->infinity) sum of f(x_i)·Δx        Δx = (b-a)/n

The area under f from a to b, as a limit of thin rectangles (a Riemann sum). ∫ is an elongated S for “sum”; dx is the strip width shrinking to zero.

The fundamental theorem of calculus

∫_a^b f(x) dx = F(b) - F(a)        where F'(x) = f(x)

To accumulate f, find an antiderivative F (a function whose rate of change is f), evaluate at the endpoints, subtract. Differentiation and integration are inverse operations.

Antiderivatives = derivative rules reversed

Integrand	Antiderivative
x^n (n != -1)	x^(n+1)/(n+1) + C
1/x (the n=-1 case)	`ln
e^x	e^x + C
sin x	-cos x + C
cos x	sin x + C

Check any of them by differentiating back.

Worked examples

Integral	Antiderivative	Result
∫_0^1 x^2 dx	[x^3/3]_0^1	1/3
∫_0^1 e^x dx	[e^x]_0^1	e - 1 ≈ 1.718
∫_0^π sin x dx	[-cos x]_0^π	1 + 1 = 2
∫_0^R 2πr dr	[πr^2]_0^R	πR^2 (closes Lesson 1)

Indefinite vs definite

• Definite ∫_a^b f dx: has limits, gives a number (the accumulated total).
• Indefinite ∫ f dx: no limits, gives a function F(x) + C (the family of antiderivatives).

The + C exists because a constant’s derivative is 0, so f has infinitely many antiderivatives differing by a constant. In a definite integral the C cancels.

Why it matters for AI

Integration is the math of continuous probability:

• density integrates to 1: ∫ f dx = 1; P(a <= X <= b) = ∫_a^b f dx
• expected value E[X] = ∫ x·f(x) dx
• entropy / KL divergence are integrals over distributions (loss functions of generative models)
• continuous-time models (neural ODEs, diffusion) solve integrals in the forward pass

Usually computed numerically (Monte Carlo, quadrature), but the FTC underwrites the theory.

Pitfalls to dodge

• Summing rectangles by hand. Use the FTC: F(b) - F(a). The Riemann sum is the definition, not the method.
• Dropping + C on an indefinite integral.
• Mishandling n = -1. ∫ 1/x dx = ln|x| + C, not a power.
• Confusing definite (number) with indefinite (function).

The one-line version

The integral is the area under a curve (a limit of thin rectangles), and the fundamental theorem computes it by reversing differentiation: find an antiderivative and subtract its endpoint values.