$\int_0^3 \lfloor x \rfloor = 3$: the floor function is nondecreasing, hence integrable, and the integral is computed from the uniform partitions

Given: $g : [0,3] \to \mathbb{R}$ with $g(x) = \lfloor x \rfloor$; a natural $M \ge 1$; $N := 3M$; and the uniform partition $U_N = (N,t)$ of $[0,3]$ with $t_i = 3\iota(i)/\iota(N) = \iota(i)/\iota(M)$ for $i \le N$ and lengths $\Delta_i = 3/\iota(N) = 1/\iota(M)$.

For every real $x$ there is exactly one integer $\lfloor x \rfloor$ with $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$ (Integer part: for every real $x$ there is exactly one integer $m$ with $m \le x < m + 1$).

$g$ is nondecreasing: for $x \le y$, $\lfloor x \rfloor \le x \le y < \lfloor y \rfloor + 1$, and $\lfloor x \rfloor$, $\lfloor y \rfloor$ are integers, so $\lfloor x \rfloor \le \lfloor y \rfloor$, no integer lying strictly between $\lfloor y \rfloor$ and $\lfloor y \rfloor + 1$ (Integer part: for every real $x$ there is exactly one integer $m$ with $m \le x < m + 1$, Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences).

A monotone function on a closed bounded interval with distinct endpoints is bounded and Riemann integrable (A monotone function on $[a,b]$ is Riemann integrable: for the uniform partition into $N$ parts the upper minus lower sum telescopes to $|f(b) - f(a)|\,(b-a)/\iota(N)$, The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$).

For $U_N$: $t_0 = 0$, $t_N = 3$, $t_i < t_{i+1}$, $\Delta_i = 1/\iota(M)$, $\sum_{i<N}\Delta_i = 3$ (Partition of $[a,b]$ as a finite strictly increasing list $a = t_0 < t_1 < \dots < t_n = b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

For a nondecreasing $g$ and a subinterval $I_i = [t_i,t_{i+1}]$: $m_i = g(t_i)$ and $M_i = g(t_{i+1})$, both attained; $L(g,P) = \sum_{i<N}m_i\Delta_i$, $U(g,P) = \sum_{i<N}M_i\Delta_i$, and $L(g,P) \le \underline{\int_0^3}g \le \overline{\int_0^3}g \le U(g,P)$ (For bounded $f$ on $[a,b]$ and a partition $P$: the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$-th subinterval, and the lower and upper Darboux sums $L(f,P) = \sum_i m_i \Delta_i$ and $U(f,P) = \sum_i M_i \Delta_i$, The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$, Greatest lower bound (infimum), Maximum and minimum of a set, Complete ordered field (least-upper-bound property)).

Finite sums: splitting a sum over $i < 3M$ into the three blocks $i < M$, $M \le i < 2M$ and $2M \le i < 3M$; scaling; and $\sum_{i<M}c = \iota(M)c$ (Finite sums and finite products, by recursion, Laws of finite sums and finite products).

For $i \in \mathbb{N}$: $\lfloor \iota(i)/\iota(M) \rfloor = 0$ when $i < M$, $= 1$ when $M \le i < 2M$, and $= 2$ when $2M \le i < 3M$; and $\lfloor \iota(j)/\iota(M)\rfloor = 3$ for $j = 3M$. Each case is [L1] applied to the displayed inequalities $0 \le \iota(i)/\iota(M) < 1$, $1 \le \iota(i)/\iota(M) < 2$, $2 \le \iota(i)/\iota(M) < 3$ and $\iota(3M)/\iota(M) = 3$, which follow from $\iota$ being strictly increasing and additive (Canonical naturals are positive and strictly increasing, The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Order is preserved by adding a constant and by adding inequalities). These order-arithmetic facts are stated by their sources for the strict order only; the nonstrict forms used here follow by adjoining the equality case, in which the two sides coincide.

For every real $\eta > 0$ there is a natural $M \ge 1$ with $1/\iota(M) < \eta$, and $\iota(M) > 0$ for $M \ge 1$ (For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$, Every complete ordered field is Archimedean, The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Canonical naturals are positive and strictly increasing).

Ordered-field arithmetic and the absolute value: adding a constant and multiplying by a positive quantity preserve an inequality; the order is total and transitive (Basic properties of the absolute value, Order is preserved by adding a constant and by adding inequalities, Sign rules for products and monotonicity of multiplication, Ordered field, Complete ordered field (least-upper-bound property)). These order-arithmetic facts are stated by their sources for the strict order only; the nonstrict forms used below follow by adjoining the equality case, in which the two sides coincide.