All Questions
Tagged with riemann-sum multivariable-calculus
18 questions
1
vote
1
answer
47
views
Double integral of two continuous functions
Let $R$ an elemental rectangle in $\mathbb{R}^{2}$ and $g, f: R \rightarrow \mathbb{R}$ continuous functions. Show that
$$
\lim _{d(P) \rightarrow 0} \sum_{i, j} f\left(u_{i j}\right) g\left(v_{i j}\...
- 58
7
votes
2
answers
852
views
Riemann vs. Darboux integrability for multiple integrals
It is well-known that for bounded functions of one variable, Darboux integrability (via upper and lower sums) is equivalent to Riemann integrability (via Riemann sums.)
I tried to generalize this to ...
- 7,206
0
votes
1
answer
28
views
Proving that $\min_{v\in S}(k\cdot f(v))=k\cdot \min_{v\in S}(f(v))$
in my textbook, while the author trying to prove that if $f$ is an integrable function on a subrectangle $S$ which is a part of a region in $\mathbb R^2$ let's call it $R$, Then we have $$\int_Rk\cdot ...
- 4,222
4
votes
1
answer
157
views
Calculate the Limit of Double Sum
Compute
\begin{equation}
L=\lim _{n \rightarrow \infty}\frac{1}{n} \sum_{a=1}^n \sum_{b=1}^n \frac{a}{a^2+b^2 }.
\end{equation}
My attempt:
Define \begin{equation} f(n,m)= \frac{1}{n} \sum_{a=1}^n \...
- 665
3
votes
1
answer
93
views
What Are the $x_i^*,y_j^*$ in the Riemann Sum Definition of the Double Integral?
the double integral $$\iint \limits_{[a,b] \times [c,d]} f(x,y) \, dxdy$$ can be represented with a Riemann Sum as $$\lim_{(\Delta x , \Delta y) \to (0,0)} \sum_{i=1}^n\sum_{j=1}^mf(x_i^*,y_j^*)\Delta ...
- 880
3
votes
1
answer
172
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Limit of Darboux sums in $\mathbb{R}^n$
Let $Q\subseteq\mathbb{R}^n$ be a rectangle $f:Q\to\mathbb{R}$ be a bounded function. Then for any $\varepsilon>0$ there exists a $\delta>0$ such that $U(f;P)\le \overline{\int}_Qf+\varepsilon$ ...
- 2,075
2
votes
1
answer
475
views
Understanding what ij mean in a Double Riemann Sum (Double Integral)
I am having trouble understanding what the ($x^*_{ij} $, $y^*_{ij}$) in this diagram (circled in blue) is explaining. What I do know is that $i$ is the iteration of the $x$ Riemann Sum and the $j$ is ...
- 125
2
votes
1
answer
839
views
Show that a function is not integrable but an iterated integral exist
So, I have the funcion
$f(x,y) =
\begin{cases}
0 & \text{if }x \text{ irrational} \\
2y & \text{if } x \text{ rational}\
\end{cases} $
Defined in $R=[0,1]\times[0,1]...
- 343
0
votes
1
answer
828
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Riemann sum of a double integral
Problem
Explain that
$$R^k_n=\sum_{i=1}^n\sum^n_{j=1}f\left(-k+\frac{2ki}{n},-k+\frac{2kj}{n}\right)\left(\frac{2k}{n}\right)^2$$
given that $$f(x,y)=\frac{1}{x^4+2x^2y^2+y^4+1}$$
is a Riemann ...
- 319
1
vote
0
answers
121
views
What does it mean for an error term to vanish "fast enough" in an integral approximation?
Let $f: [0, 1] \to \mathbf{R}$ be a non-negative, continuous function for demonstrating Riemann sums, like $f(x) = x^2$. The Riemann sum approximates the area under the curve with sums of the area of ...
- 11
1
vote
1
answer
133
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Is there an analogue to Riemann sums for double sums?
If I have a Riemann integrable function $f: [a,b] \to R$, then the following is true:
$$\frac{b-a}n \sum_{r=1}^n f(a + \frac{r(b-a)}n) \longrightarrow \int_a^b f(x)dx$$
is there an analogue for this ...
- 439
5
votes
1
answer
297
views
Show that if $\int_A f$ exists and $B$ has volume, then $\int_B f$ exists.
Let f be a real-valued function on a subset $A$ on $E^n$ and let $B \subset A$. Show that if $\int_A f$ exists and $B$ has volume, then $\int_B f$ exists.
Note: Volume is the Jordan measure. ...
- 683
0
votes
1
answer
37
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Proving $\iint_{[0,\pi]\times [0,\pi]}|\cos(x+y)|\,d(x,y)=2\pi$
$$\iint_{[0,\pi]\times[0,\pi]}\left|\cos(x+y)\right|d(x,y)=2\pi$$
I want to prove the above double integral. However, I am not sure how to open the absolute function corresponding to the proper ...
- 2,122
1
vote
0
answers
86
views
Defining a multiple integral on non-rectangular regions
Usually the Riemann integral for $\mathbb{R^n}$ is defined on a hyperrectangular region $T$, by partitioning the region's "edges", which are $(a_1,b_1) \times (a_2,b_2) \times \dots \times (a_n,b_n)$ ...
- 3,148
0
votes
1
answer
47
views
prove that $∀ε>0∃p∈P(U(f,p)−L(f,p)<ε)$
$F:[0,1]\times[0,1]\longrightarrow R$
$
f(x)=
\begin{cases}
1, & \text{y<x} \\
0, & \text{y $\geqslant$x}
\end{cases}
$
i have a problem choosing my p∈P
and proving the statement
any ...
- 321
6
votes
1
answer
1k
views
Riemann Integrability in $\Bbb R^2$
Define the General Subdivision $S$ of a rectangle $R$ in $\Bbb R^2$ as a collection $E_1,...,E_k$ of Jordan regions such that none of them has interior points in common, and:
$$R \subset \bigcup_{i=1}...
22
votes
3
answers
10k
views
General condition that Riemann and Lebesgue integrals are the same
I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
- 261
1
vote
1
answer
60
views
Limit of integral in $\mathbb R^2$
I am trying to prove the following but even if it seems easy I am not sure how to start:
Let $R$ be a rectangle in the plane and $f\colon R\rightarrow {\mathbb{R}}$ a continuous non-negative function....
- 797