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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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Riemann-Stieltjes integral problem

How can we see Riemann-Stieltjes integral geometrically? Also how defining a function $\alpha$ monotonically increasing on $[a, b]$ confirms it to be bounded? It may be silly thing to ask but I am not getting it atleast right now. Please help me.
user582661
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Problem with Definition of Riemann Integrable Function

I have difficulty understanding the definition of a Riemann integrable function $f$ on $[a,b]$. I understand the definition of upper sum $U(f,P) = \sum\limits_{i=1}^{n}M_i \Delta x_i$ and lower sum $L(f,P) = \sum\limits_{i=1}^{n}m_i \Delta x_i$,…
Rob
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Show $f(x)=x\cdot 1_{\mathbb R\setminus \mathbb Q}(x)$ is a limit of Riemann-Integrable functions?

Suppose $f(x)=\begin{cases}x&\mbox{if }x\notin Q\\0&\mbox{if }x\in Q\end{cases}$ (where Q means rational numbers, and not in Q means irrational numbers). How do you show $f(x)$ is a limit of a sequence of Riemann-Integrable functions?
user1068636
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Riemann Integrable.

We have some questions about Riemann integrable, so if $f,g$ are Riemann integrable over $[a,b]$ denote by $f,g\in R[a,b]$, then $\max\{f,g\},\min\{f,g\}\in R[a,b]$. So, it clearly ? , because in case $f
panama wannawe
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Proving the Riemann integrability of a function

Let $f : [a, b] \to \mathbb{R}$ be a Riemann integrable function. If $g : [a, b] \to \mathbb{R}$ is another function and $S = \lbrace x : f(x) \neq g(x)\rbrace$ contains exactly $n$ points, show that $g$ is also Riemann integrable from any of the…
Walls
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Given an example which shows the above theorem may not hold if $ \ g(x) <0 \ $

The first mean value theorem for integral calculus is $ \int_a^b f(x) g(x) dx=f(c) \int_a^b g(x) dx \ $ for some $ \ c \in (a,b) \ $ where $ f, g \ $ are both continuous functions and in addition $ \ g \ $ is non-negative integrable function. Given…
MAS
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Composition of two differentiable functions and a related integration and its sign

$f$ and $g$ are two differentiable functions on the real line such that $f$ is strictly increasing and $g$ is strictly decreasing. Define $p(x)=f(g(x))$ and $q(x)=g(f(x))$ for all real number $x$. Then for $t>0$ the sign of $$\int_0^t…
user514386
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Moment of inertia using double integration.

Need to find moment of inertia about $x$-axis using double integration if density at $(x,y) =y+1$ of area bounded by $x=2y-y^2$ and $x=y^2$. My difficulty arises in converting $x=2y-y^2$ to $y=?$ as one of the limits in the integral.
Danny Colautti
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Give an example of a finction $f(x)$ such that $f(x)>0$ $\forall x\in[a,b]$, where $\sqrt{f}$ is Riemann integrable but $f$ is not.

Give an example of a finction $f(x)$ such that $f(x)>0$ $\forall x\in[a,b]$, where $\sqrt{f}$ is Riemann integrable but $f$ is not. I don't know the example. Please help.
rama_ran
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By fundamental theorem of calculus, show that $f$ is odd if $\int_{-s}^sf(x)dx=0$, $\forall s\in \mathbb{R}$, $f$ being continuous function.

By fundamental theorem of calculus, show that $f$ is odd if $\int_{-s}^sf(x)dx=0$, $\forall s\in \mathbb{R}$, $f$ being continuous function. Fundamental theorem of calculus: 1st FTC: If $f$ is continuous on $[a,b]$ then the function $F$ defined…
rama_ran
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Riemann Integrability of a function and its reciprocal

I was wondering what are the conditions which need to be imposed so that the question given below holds true, Suppose $f$ is Riemann integrable on $[a,b]$ does this imply that $\frac{1}{f}$ is Riemann integrable? The condition I can think of…
physicscircus
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Riemann integrability condition

I have this function $$f(x)=\begin{cases}x^a \cos\big(\frac{1}{x}\big)&x>0\\0&x=0\end{cases}$$ defined on $[0,1]$. I found that $f$ is bounded, differentiable and continuous and for the last part I need to prove that $f$ is Riemann integrable on…
user9068702
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Riemann Integrable Function (an example)

I am supposed to give an example of a sequence of Riemann integrable functions $f_{n}$ and $f$, all defined on $[0,1]$ such that $\lim f_{n}(x)=f(x)$ but $\lim \int_{0}^{1} f_{n} (x)dx\neq\int_{0}^{1}f(x)dx$. What I can think of is…
LanaDR
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Is there a map $f:[0,1] \mapsto R$ which is unbounded?

The other condition is for any $a \in (0,1), f$ is Riemann integrable on $[a,1]$ with $\int^{1}_{a} f(x) dx =0 $.
Adienl
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Problem in Riemann integral

Show that $$\lim_{n\to \infty}\int_0^{10}\frac{n[11-x]}{1+n^2x^2}dx=11\frac{\pi}{2}$$ $[x]$ denotes the greatest integer function here $[11-x]$ is discontinuous at $0,1,2,..,10$ so i divide this integration and after solve i don't get this answer.…
RAM_3R
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