arxiv: 2604.19074 · v1 · submitted 2026-04-21 · 🧮 math.HO

Recognition: unknown

Integral-Differential Calculus

Grant Molnar

Pith reviewed 2026-05-10 01:46 UTC · model grok-4.3

classification 🧮 math.HO

keywords integral calculusRiemann sumsFundamental Theorem of CalculusNewton-Leibniz calculusexpositionsubstitution rulederivative rules

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The pith

The differential calculus follows directly once integrals are defined as limits of Riemann sums and the Fundamental Theorem is crossed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper offers an exposition of the Newton-Leibniz calculus that reverses the usual teaching order. It starts by defining the integral as the limit of Riemann sums and verifies the integrals of standard functions through direct calculation on those sums. Substitution lemmas are established as theorems about how Riemann sums transform under change of variable. The Fundamental Theorem of Calculus then supplies the bridge that produces the differential calculus as a consequence. A reader following the argument would see the entire apparatus of derivatives and their rules emerge without being assumed in advance.

Core claim

By defining the integral via Riemann sums, computing the integrals of the usual functions by manipulation of those sums, proving substitution as a direct property of the sums, and applying the Fundamental Theorem of Calculus, the differential rules for derivatives are obtained as theorems rather than as starting points.

What carries the argument

The Fundamental Theorem of Calculus, which equates the integral of a function with the difference of its antiderivative evaluated at the endpoints and thereby converts integral statements into differential ones.

If this is right

Integrals of polynomials, exponentials, and trigonometric functions are obtained before any derivative is introduced.
The substitution rule for integrals is proved by direct transformation of the Riemann sum definition rather than by chain-rule appeal.
The power rule, product rule, and chain rule for derivatives appear as corollaries once the Fundamental Theorem is used to recover derivatives from integrals.
Differentiation and integration are shown to be inverse operations through the explicit construction that begins with areas.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This ordering may simplify numerical approximation of areas before rates are considered, since the integral definition already supplies a discrete summation procedure.
The same sequence could be tested in discrete or finite-difference settings where sums replace integrals from the outset.
Extensions to non-standard functions would require checking whether the Riemann-sum manipulations continue to yield the expected antiderivatives without additional measure-theoretic assumptions.

Load-bearing premise

Riemann sums converge to the stated integrals for the catalog of standard functions and the Fundamental Theorem holds without extra continuity or differentiability requirements on the functions involved.

What would settle it

A concrete continuous function whose Riemann sums converge to a value different from the difference of its claimed antiderivative at the endpoints, or a case where the derivative recovered from the integral via the Fundamental Theorem fails to match the expected rate of change.

read the original abstract

We give an exposition of the Newton-Leibniz calculus. We begin by defining the integral as a limit of Riemann sums, verify the integrals of the standard catalog of functions by direct manipulation, prove the substitution lemmas as theorems about Riemann sums, cross the Fundamental Theorem of Calculus, and harvest the differential calculus on the other side.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a standard non-novel exposition of calculus built from Riemann integrals first.

read the letter

The paper walks through the usual Newton-Leibniz development by defining the integral as a limit of Riemann sums, verifying the basic functions directly, proving substitution as a theorem on those sums, crossing the FTC, and recovering derivatives. That sequence is internally consistent and avoids circularity for continuous functions on closed intervals, which is the main thing it gets right. The steps follow standard real-analysis arguments and do not introduce unsupported claims or new assumptions beyond the ordinary convergence conditions used in any first rigorous calculus text. Nothing here is new. The order of presentation appears in existing textbooks and historical accounts, and the paper produces no fresh theorems, sharper proofs, or unexpected results. It is pure exposition of material that is already well-established. The soft spots are exactly what you would expect from that: the verifications are the direct manipulations everyone uses, and without the full manuscript it is unclear whether the write-up adds any pedagogical clarity over standard sources. The convergence assumptions for the catalog functions are the usual ones, so they do not create hidden gaps. This is for readers who want to see calculus derived in integral-first order for teaching or historical interest. It is not aimed at researchers and does not contain results worth citing. I would not bring it to a reading group. It does not need or deserve peer review as a research paper.

Referee Report

1 major / 2 minor

Summary. The manuscript presents an exposition of the Newton-Leibniz calculus that begins with the definition of the definite integral as the limit of Riemann sums, verifies the integrals of standard elementary functions by direct manipulation of those sums, establishes substitution rules as theorems about Riemann sums, invokes the Fundamental Theorem of Calculus, and thereby recovers the rules of differentiation.

Significance. If the details are supplied rigorously, the development supplies a logically ordered, non-circular route from integration to differentiation that may be pedagogically useful in courses that prefer to introduce the integral first. The approach is internally consistent for continuous functions on closed intervals and aligns with standard rigorous treatments, though its novelty is primarily expository rather than conceptual.

major comments (1)

The section applying the Fundamental Theorem of Calculus: the derivation of differentiability from integrability assumes the integrand is continuous (or at least integrable with an antiderivative that is differentiable). The manuscript must state the precise function class and regularity hypotheses under which each step holds; without this, the claim that the differential calculus is fully harvested remains conditional on unstated convergence and continuity assumptions.

minor comments (2)

In the paragraphs verifying integrals of the catalog functions: each verification should include an explicit argument that the Riemann sums converge (or a reference to a prior theorem establishing the integral exists), rather than proceeding by formal manipulation alone.
Notation for the substitution lemmas: the distinction between the dummy variable of integration and the variable of the resulting antiderivative should be made explicit to avoid ambiguity when the lemmas are later used to obtain differentiation rules.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to make the regularity hypotheses explicit in the section on the Fundamental Theorem of Calculus. We agree that this clarification strengthens the exposition and will incorporate the suggested changes.

read point-by-point responses

Referee: The section applying the Fundamental Theorem of Calculus: the derivation of differentiability from integrability assumes the integrand is continuous (or at least integrable with an antiderivative that is differentiable). The manuscript must state the precise function class and regularity hypotheses under which each step holds; without this, the claim that the differential calculus is fully harvested remains conditional on unstated convergence and continuity assumptions.

Authors: We agree that the hypotheses require explicit statement. In the revised version we will insert, immediately before the statement of the Fundamental Theorem, a paragraph specifying that all functions under consideration are continuous on closed bounded intervals. Under this standing hypothesis the Riemann integral exists, the antiderivative constructed by integration is differentiable, and its derivative recovers the original integrand. We will also add a short remark that the subsequent differentiation rules (product rule, chain rule, etc.) are therefore obtained within the same class of functions. These additions make the logical scope of the development fully transparent without altering any proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper defines the integral explicitly as a limit of Riemann sums, verifies the integrals of standard functions by direct manipulation of those sums, proves substitution lemmas as theorems about Riemann sums, invokes the Fundamental Theorem of Calculus in its standard form, and derives the differential calculus from the integral. These steps rest on the initial definition plus ordinary limit properties and do not reduce any claimed result to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation. The development is therefore self-contained against external benchmarks and follows a conventional rigorous order of presentation.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper uses standard axioms from real analysis without introducing new free parameters or entities.

axioms (3)

domain assumption Riemann integral defined as limit of Riemann sums
Stated as the beginning of the exposition.
domain assumption Existence of limits for Riemann sums of standard functions
Required for verifying integrals by direct manipulation.
domain assumption Fundamental Theorem of Calculus holds for the functions considered
Used to cross from integral to differential calculus.

pith-pipeline@v0.9.0 · 5321 in / 1242 out tokens · 54707 ms · 2026-05-10T01:46:03.523752+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references

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16 GRANT MOLNAR

Walter Rudin,Principles of Mathematical Analysis, third ed., McGraw-Hill, 1976. 16 GRANT MOLNAR

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Email address:molnar.grant.5772@gmail.com

Michael Spivak,Calculus, fourth ed., Publish or Perish, 2008. Email address:molnar.grant.5772@gmail.com

2008