Alphabeta Math

Properties of the Integral and the Working FTC

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1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Definition

The integral with oriented limits: ∫ₐᵃ f := 0 and ∫_bᵃ f := -∫ₐᵇ f

Lemma

A function integrable on [a,b] is integrable on every closed subinterval

Theorem

Integrable functions on [a,b] form a set closed under sums and scalar multiples, and ∫ₐᵇ(λ f+μ g) = λ∫ₐᵇ f + μ∫ₐᵇ g

Theorem

If f ≤ g on [a,b] and both are integrable then ∫ₐᵇ f ≤ ∫ₐᵇ g; and m(b-a) ≤ ∫ₐᵇ f ≤ M(b-a)

Theorem

For a<c<b: f is integrable on [a,b] if and only if it is integrable on [a,c] and on [c,b], and then ∫ₐᵇ f = ∫ₐᶜ f + ∫_cᵇ f; with the oriented form for arbitrary a,b,c

Lemma

Changing an integrable function at finitely many points changes neither its integrability nor its integral

Theorem

If f is integrable on [a,b] with values in [m,M] and φ is continuous on [m,M], then φ ∘ f is integrable

Corollary

If f,g are integrable on [a,b] then so are | f|, f², fg, max(f,g) and min(f,g), and |∫ₐᵇ f| ≤ ∫ₐᵇ| f|

Theorem

If f is continuous on [a,b] and g is integrable with g ≥ 0, there is ξ ∈ [a,b] with ∫ₐᵇ fg = f(ξ)∫ₐᵇ g

Definition

The integral function F(x) := ∫ₐˣ f of an integrable f

Theorem

The integral function of a bounded integrable f is Lipschitz, hence uniformly continuous

Theorem

The first fundamental theorem: if f is integrable on [a,b] and continuous at c, then F'(c) = f(c); in particular a continuous f has F as a primitive

Theorem

The second fundamental theorem: if G is differentiable on [a,b] with G' = f and f is integrable, then ∫ₐᵇ f = G(b)-G(a)

Corollary

Every continuous function on an interval has a primitive; two primitives differ by a constant; and ∫ₐᵇ f = G(b)-G(a) for any primitive G

Theorem

If u,v are differentiable on [a,b] with u',v' integrable, then ∫ₐᵇ u v' = u(b)v(b)-u(a)v(a) - ∫ₐᵇ u'v

Theorem

Substitution: if φ is differentiable on [c,d] with φ' integrable and f is continuous on an interval containing φ([c,d]), then ∫_φ(c)^φ(d) f = ∫_cᵈ (f∘φ) φ'

Theorem

Bonnet's second mean value theorem: for f monotone and g integrable on [a,b] there is ξ∈[a,b] with ∫ₐᵇ fg = f(a)∫ₐ^ξ g + f(b)∫_ξᵇ g

Theorem

A continuous f ≥ 0 on [a,b] with ∫ₐᵇ f = 0 is identically 0

Theorem

The integral test: for f ≥ 0 nonincreasing on [0,∞), ∑ₖ f(k) converges if and only if the sequence (∫₀^N f)_N is bounded, with ∫₀^N f ≤ ∑_k<N f(k) ≤ f(0) + ∫₀^N f

Remark

Conventions of this page, and which sharpenings of the integral are taken up later in the reading order

Sources