# Questions tagged [partitions-for-integration]

For questions on finding or constructing a partition of an interval to compute the Riemann integral or Riemann-Stieltjes integral.

108 questions
Filter by
Sorted by
Tagged with
40 views

### If $f$ is Riemann integrable on a closed interval, does its integral exists for any sequence of partitions with norm converges to $0$?

It can ben shown that if $f$ is continuous on $[a, b]$ then $\int_a^bf\ dx=\lim_{n\to\infty}\ U(f, P_n)=\lim_{n\to\infty}\ L(f, P_n)$ for any sequence of partitions of $[a, b]$ with norm (or mesh) ...
• 448
22 views

• 5,354
34 views

• 1,754
21 views

### Is there a driving noise such that it behaves ''Hölderly'' over a uniform partition?

It is well-known that in case of a linear parititon of $[0,1]$, $\{t_n\}_{n=1}^N = \{\frac{n}{N}\}_{n=1}^N$, we have $$\int_{t_n}^{t_{n+1}} dt = t_{n+1} - t_n = \frac{1}{N} \quad \forall n$$ But ...
• 406
54 views

### question of partition in double integral

in the definition of double integral,what the first to do is to divide region into small subrectangles . my question is :do these partitions have to be rectangles? for example ,when dealing with polar ...
• 39
1 vote
81 views

### Riemann Sums Approximation

Let P = {$-10,-2,0,1,5$} and $f:=[-10,5]-> \mathbb{R}$ given by: $f(x)= \begin{cases} 4&\text{if}\, x= 0\\ \frac{x+2|x|}{|x|}&\text{if}\, x\not=0 \end{cases}$ I need to find a partition ...
1 vote
69 views

### Why can a partition be a refinement of itself?

Generally speaking, to call something a ‘refinement’ has certain implications. For instance, intuitively if one were to refine a partition of some interval it would make sense that the Darboux sums of ...
• 11
52 views

### A proof about a condition of Riemann integrable functions on my textbook

I was stuck when proving a theorem in Introduction to real analysis (4th edition). I don't know why the author assumed $c=x_{i}=x_{i-1}$. Is it because $x_{i}$ and $x_{i+1}$ are close enough or ...
48 views

### Defining the length of an interval in the Riemann sums

In the Riemann integration theory, the partition of the studied interval $[a,b]$ is $P(x,t)$. But how do we define the length of the subdivided intervals $[x_i , x_{i+1}]$ ? Do we say it is the ...
• 1,141
31 views

### Partitions in Definite Integral

Help me to solve it. I don't understand this knowledge!!! Thansk so much ^^ Let f(x)=|x| , [-1,1] . Have P={ x0=-1 < x1< ... <xn = 1}, xi - xi-1 = 2/n. Find: L(f,P) and U(f,P) image
33 views

• 21.5k
70 views

### When the piecewise constant integral independs of the partition's choice?

Proposition Let $I$ be a bounded interval, and let $f:I\to\textbf{R}$ be a function. Suppose that $\textbf{P}$ and $\textbf{P}'$ are partitions of $I$ such that $f$ is piecewise constant both with ...
• 21.5k
1 vote