Properties of the Integral

The integral behaves like a sum: it adds across pieces of the domain, scales and adds across functions, and preserves order. Each property is inherited from the corresponding fact about the Darboux sums that define it, and together they turn $\int_{a}^{b}$ into a linear operator on the space $R [a, b]$ of integrable functions.

Additivity over subintervals

Cutting the domain at an interior point splits the integral into two, and subdividing at a point does not affect integrability.

Additivity: the area over

[a, c]

is the area over

[a, b]

plus the area over

[b, c]

. Cutting at the interior point

b

neither creates nor destroys integrability.

Two conventions extend the formula to any ordering of the limits, so additivity holds for arbitrary $α, β, γ \in [a, b]$ :¹

b \int a f : = - a \int b f, a \int a f : = 0, α \int γ f = α \int β f + β \int γ f .

Linearity

The integral is linear: it commutes with scaling and with addition. On a fixed partition the Darboux sums already scale and split, and the operations are preserved in the passage to $sup$ and $inf$ .

Order and monotonicity

Larger integrands give larger integrals. On each subinterval $f \leq g$ forces $m_{i} (f) \leq m_{i} (g)$ , so the lower sums compare, and the ordering is preserved by the supremum.

Two consequences are used constantly. Taking $g \equiv M$ recovers the crude bound $\int_{a}^{b} f \leq M (b - a)$ ; taking $f \equiv m$ gives the lower bound. And applied to a nonnegative integrand, monotonicity says $\int_{a}^{b} f \geq 0$ whenever $f \geq 0$ .

Monotonicity of the integral. With

f \leq g

pointwise, the region under

f

sits inside the region under

g

, so

\int f \leq \int g

; the shaded band between the graphs is the nonnegative difference.

The integral triangle inequality

The size of an integral is controlled by the integral of the size. First, taking absolute values preserves integrability, because the oscillation of $∣ f ∣$ never exceeds that of $f$ .

Geometrically, $\int_{a}^{b} f$ is a signed area — lobes below the axis subtract — while $\int_{a}^{b} ∣ f ∣$ folds those lobes up and adds them, so cancellation can only shrink the magnitude.

Signed versus unsigned area. On the left

\int f

counts the lower lobe negatively, so magnitudes cancel; on the right

∣ f ∣

reflects that lobe above the axis and everything adds, giving the larger value.

Products and the algebra of integrable functions

The integrable functions are closed under multiplication, not only addition, a fact used whenever an integrand is a product.

Closure under products gives the integral form of the Cauchy–Schwarz inequality, proved exactly as in a finite sum.

The mean value theorem for integrals

A continuous function attains, somewhere on the interval, exactly its average value. Equivalently, the integral equals a rectangle whose height is a genuine function value.

The integral mean value theorem: the shaded area under

f

equals the area of a rectangle of the same width whose height

f (c)

is an attained value — the average height of

f

over

[a, b]

Two refinements matter. The first is a strict-positivity statement.

The second refinement is a weighted mean value theorem: for continuous $f$ and nonnegative integrable $w$ , some $c \in [a, b]$ gives $\int_{a}^{b} f w = f (c) \int_{a}^{b} w$ , since the average of $f$ weighted by $w$ still lies between $min f$ and $max f$ .

Invariance under finite edits

Integration ignores what happens at finitely many points.

This is why the value of an integrand at a jump (such as the $\frac{1}{2}$ at the discontinuity of a step function) never affects its integral.

Property	Statement	Source of the proof
Additivity	$\int_{a}^{c} f = \int_{a}^{b} f + \int_{b}^{c} f$	partitions split at $b$
Linearity	$\int (α f + β g) = α \int f + β \int g$	sums scale and add
Monotonicity	$f \leq g \Rightarrow \int f \leq \int g$	$m_{i} (f) \leq m_{i} (g)$
Triangle	$∣ \int f ∣ \leq \int ∣ f ∣$	$- ∣ f ∣ \leq f \leq ∣ f ∣$
Products	$f g \in R [a, b]$	polarization of squares
Mean value	$\int_{a}^{b} f = f (c) (b - a)$	bounds plus IVT

Lebl, Basic Analysis I, §5.2 — additivity (Lemma 5.2.1, Prop. 5.2.2), linearity (Prop. 5.2.4), monotonicity (Prop. 5.2.6), and invariance under changing finitely many values (Prop. 5.2.10), together with the $\int_{b}^{a} = - \int_{a}^{b}$ conventions. ↩ ↩² ↩³ ↩⁴ ↩⁵
Shkoller, MAT125B Lecture Notes, §1.8 — linearity of the Riemann integral (Thm. 1.22), reduced to the linearity $S_{δ} (f + g) = S_{δ} (f) + S_{δ} (g)$ of Riemann sums. ↩
Shkoller, MAT125B Lecture Notes, §1.9 — integrability of $∣ f ∣$ and the bound $∣ \int f ∣ \leq \int ∣ f ∣$ (Thm. 1.30), via $U_{δ} (∣ f ∣) - L_{δ} (∣ f ∣) \leq U_{δ} (f) - L_{δ} (f)$ . ↩
Shkoller, MAT125B Lecture Notes, §1.13 (Problems 1.2–1.3) — $f \in R (a, b) \Rightarrow f^{2} \in R (a, b)$ and the polarization identity $(f - g)^{2} - (f + g)^{2} = - 4 f g$ giving $f g \in R (a, b)$ . ↩
Shkoller, MAT125B Lecture Notes, §1.9 — the intermediate value (mean value) theorem for integrals (Thm. 1.28): a continuous $f$ attains its average value; cf. Lebl §5.2 Exercise 5.2.4. ↩

Additivity over subintervals

Linearity

Order and monotonicity

The integral triangle inequality

Products and the algebra of integrable functions

The mean value theorem for integrals

Invariance under finite edits

Footnotes