Proof of the second fundamental theorem of calculus

Builds on Riemann integrable function Riemann integral Riemann sum Mesh of a partition Partition of an interval The reals are Archimedean Mean value theorem

Proof

We show that $F(b) - F(a)$ is a Riemann sum of $f$ for every partition.

Let $a < b$ , let $F$ be continuous on $[a, b]$ and differentiable on $(a, b)$ , and let $f \colon [a, b] \to \mathbb{R}$ be Riemann integrable with $f = F'$ on $(a, b)$ . Let $I = \int_a^b f(x) \, dx$ be its integral.
Let $P = (x_0, \dots, x_n)$ be a partition of $[a, b]$ . The sum telescopes: $F(b) - F(a) = \sum_{i=1}^{n} \big(F(x_i) - F(x_{i-1})\big)$ .
Fix $i$ . The function $F$ is continuous on $[x_{i-1}, x_i]$ and differentiable on $(x_{i-1}, x_i) \subseteq (a, b)$ . By the mean value theorem, there is $c_i \in (x_{i-1}, x_i)$ with $F(x_i) - F(x_{i-1}) = F'(c_i)(x_i - x_{i-1}) = f(c_i)(x_i - x_{i-1})$ .
The points $c_i$ are tags for $P$ . By steps 2 and 3, $F(b) - F(a) = S(f, P, c)$ , the Riemann sum of $f$ for $(P, c)$ .
Let $\varepsilon > 0$ . By integrability, there is $\delta > 0$ with $|S(f, P, t) - I| < \varepsilon$ for every tagged partition of mesh less than $\delta$ .
By the Archimedean property, take $n$ with $n \delta > b - a$ , and the partition $P$ with points $a + i (b - a)/n$ . Its mesh is less than $\delta$ .
With the tags $c$ of step 3 for this $P$ , steps 4 and 5 give $|F(b) - F(a) - I| < \varepsilon$ .
This holds for each $\varepsilon > 0$ . So $\int_a^b f(x) \, dx = F(b) - F(a)$ .

Idea

On each piece of a partition, the mean value theorem turns the increase of $F$ into $f$ at some point times the length of the piece. So the total increase $F(b) - F(a)$ is exactly a Riemann sum of $f$ , for every partition. Fine partitions make Riemann sums close to the integral, so the two numbers agree.

Reading path

To understand this page from the ground up, read these 141 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof of the epsilon-delta form of continuity, Proof that there is a natural number n with 1/n below any positive epsilon, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the linearity of differentiation, Proof of the sequential characterization of limits, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the mean value theorem from Rolle's theorem, Proof of the constant multiple rule.

Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Integers object
Limit point of a set definition
Local maximum definition
Mathematical induction concept
Natural numbers object
Ordered field definition
Partition of an interval definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Set definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Archimedean ordered field definition
Successor function definition
Mesh of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Ordered pair definition
Continuity at a point definition
Derivative definition
Addition of natural numbers definition
Proof of the Archimedean property in an Archimedean ordered field proof
Riemann integrable function definition
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Truth table concept
Modus ponens concept
Universal quantifier definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Completeness axiom statement
Cartesian product definition
Continuous function definition
Differentiable function definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
Riemann integral definition
Subset definition
The rationals form an ordered field statement
Propositional logic concept
Mathematical proof concept
Convergent sequence definition
Sequential continuity definition
Binary relation definition
Order on the reals definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Tautology definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Interval definition
Strong induction statement
Logical equivalence definition
Preorder definition
Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Epsilon-delta definition of continuity statement
Epsilon-delta definition of a limit statement
Well-ordering principle statement
Double negation law statement
Total order definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Triangle inequality statement
Absolute value is multiplicative statement
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
The reals are Archimedean statement
Proof that convergent sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Proof that there is a natural number n with 1/n below any positive epsilon proof
Limit of a sum of sequences statement
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Squeeze theorem statement
Convergent sequences are bounded statement
Monotone convergence theorem for sequences statement
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
Proof that a decreasing sequence bounded below converges proof
Decreasing sequence bounded below converges statement
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Limit of a product of sequences statement
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the linearity of differentiation proof
Bolzano–Weierstrass theorem statement
Linearity of differentiation statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the constant multiple rule proof
Boundedness theorem statement
Constant multiple rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Extreme value theorem statement
Proof of Fermat's theorem on stationary points proof
Fermat's theorem on stationary points statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem statement
Proof of the second fundamental theorem of calculus proof

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628 pages build on this one.

Tag	`P03J7CNQPJ`
URI	`m2://P03J7CNQPJ`
Kind	proof
Writing	W0 stub
MSC	26A42
Proves	Second fundamental theorem of calculus
Uses	Riemann integrable function, Riemann integral, Riemann sum, Mesh of a partition, Partition of an interval, The reals are Archimedean, Mean value theorem
Changed	4 November 2026

Proof of the second fundamental theorem of calculus

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