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Questions tagged [riemann-integration]

The Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.

Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be Riemann-integrable. The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough.

The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral.

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How much do we really care about Riemann integration compared to Lebesgue integration?

Let me ask right at the start: what is Riemann integration really used for? As far as I'm aware, we use Lebesgue integration in: probability theory theory of PDE's Fourier transforms and really, anywhere I can think of where integration is used…
Ennar
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Is Dirichlet function Riemann integrable?

"Dirichlet function" is meant to be the characteristic function of rational numbers on $[a,b]\subset\mathbb{R}$. On one hand, a function on $[a,b]$ is Riemann integrable if and only if it is bounded and continuous almost everywhere, which the…
Vladimir
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What's the "limit" in the definition of Riemann integrals?

Consider one of the standard methods used for defining the Riemann integrals: Suppose $\sigma$ denotes any subdivision $a=x_0
user9464
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Why does the monotone convergence theorem not apply on Riemann integrals?

I had just learned in measure theory class about the monotone convergence theorem in this version: For every monotonically increasing sequence of functions $f_n$ from measurable space $X$ to $[0, \infty]$, $$ \text{if}\quad \lim_{n\to \infty}f_n =…
Amit Keinan
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Integrability of Thomae's Function on $[0,1]$.

Consider the function $f: [0,1] \to \mathbb{R}$ where $f(x)=$ \begin{cases} \frac 1q & \text{if } x\in \mathbb{Q} \text{ and } x=\frac pq \text{ in lowest terms}\\ 0 & \text{otherwise} \end{cases} Determine whether or not $g$ is in $\mathscr{R}$ on…
Anthony Peter
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Show that $\int_0^1 \left(\left\lfloor\frac{\alpha}{x}\right\rfloor-\alpha\left\lfloor\frac{1}{x}\right\rfloor\right)\mathrm dx=\alpha \ln\alpha$

Show that the improper integral $\int_0^1 \left(\left\lfloor\frac{\alpha}{x}\right\rfloor-\alpha\left\lfloor\frac{1}{x}\right\rfloor\right)\mathrm dx=\alpha \ln\alpha$, for $\alpha\in(0,1)$. This is an integral of Riemann. My work: The set of…
user173262
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The absolute value of a Riemann integrable function is Riemann integrable.

This is an exercise in Bartle & Sherbert's Introduction to Real Analysis second edition. They ask to show that if $I=[a,b]$ is a closed bounded interval and that $f:I\to\mathbb{R}$ is (Riemann) integrable on $I$, then $|f|$ is integrable on $I$. Of…
Spenser
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How to decide whether Lebesgue integral or Riemann integral?

Very often I feel very uncomfortable in dealing with integrals, since I am wondering whether the given integral is meant as a (improper) Riemann integral or Lebesgue integral? For instance, the Gamma function is often defined by the Euler…
Hasti Musti
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Let $g$ be a Riemann integrable function on $[a,b]$, and $f$ is a continuous. Prove that $f(g(x))$ is Riemann integrable for all $x\in[a,b]$.

Let $g$ be a Riemann integrable function on $[a,b]$, and assume $f$ is a continuous function defined on $g(x)$, for all $x\in[a,b]$. Prove that $f(g(x))$ is Riemann integrable for all $x\in[a,b]$. What I have so far: I know that if $f(g)$ is…
secondaccount
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What does it mean for a function to be Riemann integrable?

I need to come up with a precise mathematical definition of what a Riemann integrable function is. I know what the Riemann integral is but when I look for definitions all I find are proofs of how to prove that a function is Riemann integrable. I…
idknuttin
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What is the difference between Riemann and Riemann-Stieltjes integrals?

I'm quite confused, what is the difference between these two integrals (R and RS)? It seems that RS is closer to Lebesgue in its treatment of discontinuities, but otherwise I don't understand. If someone could give an example of a function for which…
user19821
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Building intuition for Riemann-Stieltjes integral

Soft question: how can I build geometric intuition for and/or visualize the Riemann–Stieltjes integral? This is pretty easy with Riemann-integral, but in case of Riemann-Stieltjes integral I always have to rely on symbolics.
qarabala
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Calculate the following integral

$$\int_{[0,1]^n} \max(x_1,\ldots,x_n) \, dx_1\cdots dx_n$$ My work: I know that because all $x_k$ are symmetrical I can assume that $1\geq x_1 \geq \cdots \geq x_n\geq 0$ and multiply the answer by $n!$ so we get that $\max(x_1\ldots,x_n)=x_1$ and…
lfc
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Convergence of Riemann sums for improper integrals

I was considering whether or not the limit of Riemann sums converges to the value of an improper integral on a bounded interval. This appears to be true in some cases when the sum avoids points where the function is not defined. For example, the…
WoodWorker
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What are necessary and sufficient conditions for Riemann integrability?

Michael Spivak, in his "Calculus" writes Although it is possible to say precisely which functions are integrable,the criterion for integrability is too difficult to be stated here I request someone to please state that condition.Thank you very…
user31029
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