${\int }_0^3\lfloor x\rfloor =3$: 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 $t_i=3\iota (i)\mathrm{/}\iota (N)=\iota (i)\mathrm{/}\iota (M)$ for $i\le N$ and lengths ${\mathrm{\Delta }}_i=3\mathrm{/}\iota (N)=1\mathrm{/}\iota (M)$.

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 $\mathrm{\mid }f(b)-f(a)\mathrm{\mid }(b-a)\mathrm{/}\iota (N)$, 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 $U_N$: $t_0=0$, $t_N=3$, $t_i<t_{i+1}$, ${\mathrm{\Delta }}_i=1\mathrm{/}\iota (M)$, ${\sum }_{i<N}{\mathrm{\Delta }}_i=3$ (Partition of $[a,b]$ as a finite strictly increasing list $a=t_0<t_1<\cdots <t_n=b$, 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; $L(g,P)={\sum }_{i<N}{m}_i{\mathrm{\Delta }}_i$, $U(g,P)={\sum }_{i<N}{M}_i{\mathrm{\Delta }}_i$, and $L(g,P)\le \underset{‾}{{\int }_0^3}g\le \overline{{\int }_0^3}g\le U(g,P)$ (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 $L(f,P)={\sum }_i{m}_i{\mathrm{\Delta }}_i$ and $U(f,P)={\sum }_i{M}_i{\mathrm{\Delta }}_i$, 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 $i\in \mathbb{N}$: $\lfloor \iota (i)\mathrm{/}\iota (M)\rfloor =0$ when $i<M$, $=1$ when $M\le i<2M$, and $=2$ when $2M\le i<3M$; and $\lfloor \iota (j)\mathrm{/}\iota (M)\rfloor =3$ for $j=3M$. Each case is [L1] applied to the displayed inequalities $0\le \iota (i)\mathrm{/}\iota (M)<1$, $1\le \iota (i)\mathrm{/}\iota (M)<2$, $2\le \iota (i)\mathrm{/}\iota (M)<3$ and $\iota (3M)\mathrm{/}\iota (M)=3$, 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 $1\mathrm{/}\iota (M)<\eta$, and $\iota (M)>0$ for $M\ge 1$ (For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$, 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.