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Riemann integration defines an integral by requiring that every sufficiently fine tagged Riemann sum approximate the same number. For a bounded function $f:[a,b]\to \mathbb{R}$, a partition of an interval $a=x_0<\cdots <x_n=b$ and tags ${\xi }_i\in [x_{i-1},x_i]$ giveThe mesh of a partition must tend to zero, and the answer must be independent of the tags and the chosen partitions of an interval. The equivalent Darboux sum formulation traps every Riemann sum between a lower Darboux sum and an upper Darboux sum. Their common limiting value, when it exists, is the Riemann integral.^[ref]

For $f(x)=x$ on $[0,1]$, equal-width partitions of an interval give lower and upper values $1/2\mp 1/(2n)$, so the Riemann integral is $1/2$. For the Dirichlet function, every nondegenerate subinterval has infimum zero and supremum one: lower and upper values remain zero and one. Refining the partition of an interval cannot resolve that gap. Thus boundedness alone does not ensure Riemann integrability; the oscillation of a function on an interval must become small in the weighted aggregate.

For a continuous function $f:[a,b]\to \mathbb{R}$ and an integrator $g$ of bounded variation, the Riemann-Stieltjes integral is the common limit, as the mesh of a partition tends to zero, ofUnlike an ordinary Riemann sum, it weights each sample by the change in $g$, rather than by the interval length. Uniform continuity of $f$ and the finite total variation of a function bound make the tagged sums Cauchy: passing to a common partition refinement changes a sum by at most the modulus of continuity of $f$ at its mesh times the total variation of a function of $g$.

If $g(x)=x$, this is the ordinary Riemann integral. If $g$ has a single unit jump at an interior point $c$ and is otherwise constant, the value is $f(c)$: all its weight concentrates at that point. For a continuous nondecreasing $g$, the associated Lebesgue-Stieltjes measure, with no atom at $a$, satisfies $\mu ((u,v])=g(v)-g(u)$ and gives the same integral. More general discontinuous integrators require attention to endpoint and one-sided conventions when identifying their signed measures.

Vector-valued integrators are treated componentwise. In particular, a continuous path of bounded variation permits the iterated Riemann-Stieltjes integrals defining its path signature.

For a bounded function on a fixed closed interval, Riemann integrability with Riemann integral $I$ is equivalent to convergence of all tagged Riemann sums to $I$ as the mesh of a partition tends to zero. Precisely, for every $\epsilon >0$ there is $\delta >0$ such that every partition of an interval of mesh less than $\delta$, with any tags in the corresponding closed subintervals, has its Riemann sum within $\epsilon$ of $I$.

A right Riemann sum on a partition $a=x_0<\cdots <x_N=b$ samples each subinterval at its right endpoint. For a continuous function its limit as the mesh tends to zero is the integral, by the Riemann sum convergence theorem. On a uniform partition of width $h$, comparison of endpoint weights in the composite trapezoidal rule gives $R_N-T_N=h(f(b)-f(a))/2$. Thus $N^{-5}{\sum }_{j=1}^N{j}^4$ is a right Riemann sum for ${\int }_0^1{x}^{4}dx$, and its limit is $1/5$.

For fixed $\alpha \in [0,1)$, design $x_i=i/n$, $m\in C^2[0,1]$, a nonzero nonnegative symmetric piecewise $C^1$ regression kernel supported on $[-1,1]$, and $h\to 0$, $n{h}^2\to \infty$, write ${\mu }_{r,\alpha }={\int }_{-\alpha }^1{u}^{r}K(u)du$ and ${\beta }_r={\mu }_{r,\alpha }/{\mu }_{0,\alpha }$. On the rescaled grid of spacing $1/(nh)$, the Riemann sum error for $u^{r}K(u)$ is $O((nh{)}^{-1})$, by bounded variation. The normalized first and second local moments are therefore $h{\beta }_1+O(n^{-1})$ and $h^2{\beta }_2+O(h/n)$. A uniform second-order Taylor expansion gives the displayed bias of an estimator, since $n^{-1}=o(h^2)$. Equivalently it is $h{\beta }_1{m}^{\prime }(0)+h^2(\alpha {\beta }_1+{\beta }_2/2)m^{\prime \prime }(0)+o(h^2)$. The first-order term explains the generic boundary bias of a local constant estimator.

Aligned two-dimensional sections constrain the three-dimensional structure of a specimen. For parallel sections of known spacing $\Delta z$, the Riemann sum ${\sum }_i{A}_i\Delta z$ approximates volume. Distortion, missing sections and alignment errors constrain the reconstruction; the volume estimate alone does not determine the identity or developmental fate of each visible structure.

On a uniform partition of $[a,b]$ with width $h$, the right Riemann sum and the composite trapezoidal rule have equal weights at every interior mesh point. They differ only by adding half a final-endpoint weight and removing half an initial-endpoint weight. This proves the displayed identity. For continuous $f$, both converge to its definite integral; their finite endpoint correction need not be confused with the smaller trapezoidal discretization error for a smooth integrand.

For the transmitting product with factors $\cosh [(\theta -i\pi j/N)/2]/\cosh [(\theta +i\pi j/N)/2]$ and prefactor $(-1{)}^N$, the continuous phase on $\theta \ge 0$ is $\pi N-2{\sum }_{j=1}^{N-1}\arctan [\tanh (\theta /2)\tan (\pi j/(2N))]$. Its derivative is $-{\sum }_j\sin (\pi j/N)/[\cosh \theta +\cos (\pi j/N)]$. A Riemann sum at fixed positive rapidity gives $(2N/\pi )\log \tanh (\theta /2)$ at leading order. The endpoints depend on the stated unwrapped branch; setting the high-energy value to zero silently changes the constant.