Riemann Integral

Checkpoints

• Additivity of the Integral Basis If f is Riemann integrable on [a, b] and c ∈ (a, b), then f is integrable on [a, c] and [c, b], and the integrals satisfy ∫_a^b f = ∫_a^c f + ∫_c^b f. This checkpoint proves additivity from Darboux sums, extends it consistently to oriented intervals via the convention ∫_a^b f = −∫_b^a f, and uses it to introduce the integral with variable upper limit F(x) = ∫_a^x f(t) dt — the bridge to the Newton–Leibniz formula.
• Change of Variables in the Integral Basis If φ is a continuously differentiable function on [α, β] with φ([α, β]) ⊆ [a, b] and f is continuous on [a, b], then ∫_α^β f(φ(t)) φ'(t) dt = ∫_{φ(α)}^{φ(β)} f(x) dx. This checkpoint derives the substitution rule from the chain rule plus Newton–Leibniz, treats both the definite-integral form (with limits transformed) and the indefinite-integral form ∫ f(φ(t)) φ'(t) dt = ∫ f(x) dx, and walks through the standard substitution patterns — linear, trigonometric, and inverse.
• Definition of the Riemann Integral Basis The Riemann integral of a bounded function on [a, b] is defined as the common limit of upper and lower Darboux sums over progressively finer partitions, when those limits agree. This checkpoint sets up partitions, upper and lower sums, and Riemann integrability, and shows that every continuous function on a closed interval is integrable.
• Convergence of Improper Integrals Basis Convergence tests for improper integrals mirror those for infinite series. This checkpoint develops the comparison test, the limit comparison test, the Cauchy convergence criterion, and the Abel–Dirichlet tests for oscillatory integrals such as ∫_1^∞ (sin x) / x dx; defines absolute versus conditional convergence; and uses the second mean value theorem to handle the trickier oscillatory cases.
• Improper Integrals Basis The Riemann integral is defined on bounded intervals for bounded functions; improper integrals extend it to unbounded intervals (∫_a^∞ f) and to functions unbounded near an endpoint (∫_a^b f with f blowing up at b) by taking a limit of proper integrals. This checkpoint defines both kinds of improper integrals, distinguishes convergence from divergence, and works through canonical examples: ∫_1^∞ x^(−p) dx, ∫_0^1 x^(−p) dx, and the gamma integral ∫_0^∞ e^(−x) x^(s−1) dx.
• Integration by Parts Basis Integrating the product rule (uv)' = u'v + uv' over [a, b] and applying Newton–Leibniz gives the integration-by-parts formula ∫_a^b u(x) v'(x) dx = [u(x) v(x)]_a^b − ∫_a^b u'(x) v(x) dx. This checkpoint derives the formula, presents the indefinite-integral form ∫ u dv = uv − ∫ v du, and works through canonical applications: ∫ x e^x dx, ∫ ln x dx, and the reduction formula for ∫ x^n e^x dx.
• Mean Value Theorem for Integrals Basis If f is continuous on [a, b], there exists ξ ∈ [a, b] with ∫_a^b f(x) dx = f(ξ) (b − a) — the integral equals the value at some intermediate point times the interval length. This checkpoint proves the theorem by combining the monotonicity bound m(b − a) ≤ ∫ f ≤ M(b − a) with the Intermediate Value Theorem, presents the weighted form ∫ f g = f(ξ) ∫ g for non-negative weight g, and interprets the result geometrically as 'average value of f on [a, b]'.
• Monotonicity of the Integral Basis If f, g are integrable on [a, b] and f ≤ g pointwise, then ∫_a^b f ≤ ∫_a^b g. This checkpoint proves monotonicity from the definition of Darboux sums, derives the standard estimate |∫_a^b f| ≤ ∫_a^b |f|, and obtains the bound m(b − a) ≤ ∫_a^b f ≤ M(b − a) for any bounds m ≤ f ≤ M — the ingredients of the integral mean value theorem.
• The Newton–Leibniz Formula Basis The fundamental theorem of calculus has two halves: for f continuous on [a, b], the function F(x) = ∫_a^x f(t) dt is differentiable with F'(x) = f(x); and for any primitive G of f, ∫_a^b f(x) dx = G(b) − G(a). This checkpoint proves both halves — the first by applying the integral mean value theorem to the difference quotient, the second by combining the first with the 'primitives differ by a constant' lemma — and shows how Newton–Leibniz reduces evaluating integrals to finding antiderivatives.
• Primitives (Antiderivatives) Basis A primitive of f on an interval I is a differentiable function F with F'(x) = f(x) for every x ∈ I; any two primitives of the same f differ by a constant. This checkpoint defines the primitive, proves the 'differs by a constant' theorem from Lagrange's mean value theorem, introduces the indefinite-integral notation ∫ f(x) dx = F(x) + C, and explains the difference in nature between the indefinite integral (a family of functions) and the definite integral (a number).
• Classes of Riemann-Integrable Functions Basis Beyond continuous functions, two large classes of bounded functions on a closed interval are Riemann integrable: monotonic functions, and functions with only finitely many discontinuities (more generally, with discontinuities forming a set of measure zero — Lebesgue's criterion). This checkpoint proves integrability of monotonic and piecewise-continuous functions, states Lebesgue's characterisation in terms of the set of discontinuities, and exhibits a bounded function (the Dirichlet function) that fails to be Riemann integrable.
• Second Mean Value Theorem for Integrals Basis If g is monotonic on [a, b] and f is integrable, there exists ξ ∈ [a, b] with ∫_a^b f(x) g(x) dx = g(a) ∫_a^ξ f(x) dx + g(b) ∫_ξ^b f(x) dx — Bonnet's form of the second mean value theorem. This checkpoint derives the theorem by an Abel-summation argument that pairs naturally with integration by parts, presents the special cases for g non-negative monotone, and uses it to establish convergence of oscillatory integrals such as ∫_1^∞ (sin x) / x dx.