Cheatsheet: Why area equals slope

The puzzle

Why does ∫_a^b f dx = F(b) - F(a)? Why does an area equal a difference of values of a related function? Area and slope look unrelated. They are not.

The area function

A(x) = ∫_a^x f(t) dt        (area under f, from fixed a to moving x)

The one geometric move

Slide the right end by dx. The new area is a thin sliver: width dx, height f(x).

A(x + dx) - A(x) ≈ f(x) · dx
[ A(x+dx) - A(x) ] / dx ≈ f(x)
as dx -> 0:   A'(x) = f(x)

The derivative of the area function is the original curve.

That is the fundamental theorem

A' = f means A is an antiderivative of f. Any antiderivative F differs by a constant (which cancels), so:

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

Integration and differentiation are inverse operations because the slope of an accumulated area is the thing being accumulated.

Worked examples (A’(x) = f(x) every time)

f	area function A(x)	A'(x)
x^2	x^3/3	x^2
e^x	e^x - 1	e^x
2πr (circle)	πR^2	2πR

The circle case is L1’s observation, now proved general.

The everyday version

A bucket under a tap: the rate the water level rises equals the tap’s flow at that moment. Water level = the integral; flow = the function; fill-rate = flow is A'(x) = f(x). The rate a total accumulates is the thing accumulating.

Why it matters for AI

The PDF-and-CDF pairing: the cumulative distribution F(x) = ∫ f(t) dt has derivative F'(x) = f(x), the density. Every PDF/CDF pair is A' = f. Same with any running total vs its rate: cumulative loss whose slope is current loss, cumulative count whose slope is the instantaneous rate.

Pitfalls to dodge

• Treating the FTC as a coincidence. It follows from the sliver: extending area by dx adds f(x)·dx.
• Forgetting the start point. A is measured from a fixed a; different starts differ by a constant (the +C).
• Confusing the curve with its area function. f is the height (rate); A is the accumulated area (total).
• Thinking area and slope are unrelated. The slope of the area function is the curve.

The one-line version

The slope of an accumulated area is the curve being accumulated, so integration and differentiation undo each other, which is the fundamental theorem.