Riemann Integral

The Riemann integral formalizes the notion of "area under a curve" by approximating it with rectangular sums. A function is Riemann integrable if upper and lower sums converge to the same value. Every continuous function on $[a, b]$ is Riemann integrable, as are bounded functions with finitely many discontinuities. The Riemann integral is the foundation for calculus and provides a rigorous definition of integration.

Partitions and Riemann sums

Definition6.1Partition

A partition of $[a, b]$ is a finite sequence $P = \{x_0, x_1, \ldots, x_n\}$ with

$a = x_0 < x_1 < \cdots < x_n = b.$

The mesh (or norm) of $P$ is $\|P\| = \max_{1 \leq i \leq n} (x_i - x_{i-1})$ .

Definition6.2Riemann sum

For a partition $P$ and sample points $c_i \in [x_{i-1}, x_i]$ , the Riemann sum is

$S(f, P, \{c_i\}) = \sum_{i=1}^n f(c_i)(x_i - x_{i-1}).$

RemarkUpper and lower sums

The upper sum $U(f, P) = \sum_{i=1}^n M_i (x_i - x_{i-1})$ uses $M_i = \sup_{[x_{i-1}, x_i]} f$ . The lower sum $L(f, P) = \sum_{i=1}^n m_i (x_i - x_{i-1})$ uses $m_i = \inf_{[x_{i-1}, x_i]} f$ .

ExampleRiemann sum for a step function

Let $f(x) = 1$ for $x \in [0, 1]$ . For the partition $P = \{0, 1/n, 2/n, \ldots, n/n\}$ , the Riemann sum (with any sample points) is

$S(f, P) = \sum_{i=1}^n 1 \cdot \frac{1}{n} = 1.$

As $n \to \infty$ , all Riemann sums converge to $1$ , so $\int_0^1 1 \, dx = 1$ .

Definition of Riemann integrability

Definition6.3Riemann integrability

A bounded function $f : [a, b] \to \mathbb{R}$ is Riemann integrable if

$\sup_P L(f, P) = \inf_P U(f, P).$

This common value is the Riemann integral, denoted $\int_a^b f(x) \, dx$ or $\int_a^b f$ .

RemarkEquivalent definition

$f$ is Riemann integrable if and only if for every $\epsilon > 0$ , there exists a partition $P$ such that $U(f, P) - L(f, P) < \epsilon$ .

ExampleContinuous functions are Riemann integrable

If $f : [a, b] \to \mathbb{R}$ is continuous, then $f$ is Riemann integrable. Proof: by uniform continuity (via Heine-Cantor), for any $\epsilon > 0$ , there exists $\delta$ such that $|f(x) - f(y)| < \epsilon/(b-a)$ whenever $|x - y| < \delta$ . For a partition with mesh $< \delta$ , $M_i - m_i < \epsilon/(b-a)$ , so $U(f, P) - L(f, P) < \epsilon$ .

ExampleStep functions are integrable

A step function $f$ (constant on finitely many intervals) is Riemann integrable. Choose a partition aligning with the jump points; then $U(f, P) = L(f, P)$ .

ExampleDirichlet function is not Riemann integrable

The Dirichlet function

$f(x) = \begin{cases} 1 & x \in \mathbb{Q}, \\ 0 & x \notin \mathbb{Q} \end{cases}$

on $[0, 1]$ is not Riemann integrable. For any partition $P$ , $M_i = 1$ and $m_i = 0$ on every subinterval (by density of rationals and irrationals), so $U(f, P) = 1$ and $L(f, P) = 0$ . Thus $\sup L(f, P) = 0 \neq 1 = \inf U(f, P)$ .

Properties of the Riemann integral

Theorem6.1Linearity

If $f, g$ are Riemann integrable on $[a, b]$ and $\alpha, \beta \in \mathbb{R}$ , then

$\int_a^b (\alpha f + \beta g) = \alpha \int_a^b f + \beta \int_a^b g.$

Theorem6.2Monotonicity

If $f \leq g$ on $[a, b]$ , then $\int_a^b f \leq \int_a^b g$ .

Theorem6.3Triangle inequality

$\left|\int_a^b f\right| \leq \int_a^b |f|.$

Fundamental Theorem of Calculus

Theorem6.4Fundamental Theorem of Calculus (Part I)

If $f$ is continuous on $[a, b]$ , then $F(x) = \int_a^x f(t) \, dt$ is differentiable on $(a, b)$ with $F'(x) = f(x)$ .

Theorem6.5Fundamental Theorem of Calculus (Part II)

If $f$ is continuous on $[a, b]$ and $F' = f$ , then

$\int_a^b f(x) \, dx = F(b) - F(a).$

ExampleComputing ∫₀¹ x² dx

Let $f(x) = x^2$ . Then $F(x) = x^3/3$ is an antiderivative. By FTC Part II,

$\int_0^1 x^2 \, dx = F(1) - F(0) = \frac{1}{3} - 0 = \frac{1}{3}.$

Summary

The Riemann integral formalizes integration via partitions and limits:

Defined using upper and lower sums.
Continuous functions are Riemann integrable.
Properties: linearity, monotonicity, triangle inequality.
Fundamental Theorem of Calculus connects integration and differentiation.

See Integrability Criterion and Fundamental Theorem for proofs.