All Questions
48
questions
0
votes
1
answer
46
views
Prove $(x,y)\rightarrow f(x,y)=\dfrac {\cos(xy)}{(1+x^2)(1+y)}\in L^{1}(]0,+\infty×]-1,1[)$
today I have a question about prove this function
$$(x,y)\rightarrow f(x,y)=\dfrac {\cos(xy)}{(1+x^2)(1+y)}$$
Such that $(x,y)\in ]0,+\infty ×]-1,1[$ are integrable.
**My attempts **
I was use this ...
- 2,323
5
votes
1
answer
405
views
How to show that two integrals are equal?
In my advanced calculus course, I am struggling with the following problem where we need to show that the two integrals are equal.
Consider a function $g:[0,1] \times [0,1] \to \mathbb{R}$
defined by
$...
- 159
0
votes
0
answers
102
views
Showing $\frac{1}{x \ln (x)}$ is in $L^p$ for all $p>0$
I was trying to find a function which is in $L^p[0,1]$ for all $p$ but not in $L^{\infty}[0,1]$. I asked my professor, when he said it's $\frac{1}{x \ln (x)}$. However, I am having a hard time to show ...
- 420
0
votes
2
answers
119
views
An upper bound for an integral
I saw many references using the following estimate but I couldn't prove it.
Given $T>0$ and $0 < b \leq \frac{1}{2}$, exist $C(b)$ constant that depends only on $b$ such that
\begin{equation}
\...
- 119
0
votes
1
answer
37
views
Calculate $S_2(\varphi(U))$
Hey I have a question for this problem:
Let $U := (0, 4) × (0, 2\pi) ⊆ \mathbb{R}^2$ and $\varphi : U → \mathbb{R}^3$ given by $\varphi(r, θ) := (r \cos θ, r \sin θ, r)$.
Calculate $S_2(\varphi(U))$. ...
- 900
0
votes
1
answer
47
views
Exercise on integration by substitution
So I have this exercise:
A is defined as $A := \{ (x,y) \in \mathbb{R}^2: 0<y<x \text{ and } 0<x+y<1\}$
We must show that $φ : A → \{(s, t) ∈ \mathbb{R}^2 : 0 < s < t < 1\}$ $φ((...
- 900
1
vote
1
answer
34
views
Calculate $\int_{A}(x + y)^{1/3} (x − y)^{1/2} dλ_2(x, y)$ using substitution
So I have this exercise:
A is defined as $A := \{ (x,y) \in \mathbb{R}^2: 0<y<x \text{ and } 0<x+y<1\}$
We must show that $φ : A → \{(s, t) ∈ \mathbb{R}^2 : 0 < s < t < 1\}$ $φ((...
- 900
1
vote
2
answers
44
views
Calculate the volume $λ_3(A)$ of $A := \{ (x, y, z) ∈\mathbb{R}^3 : x^2 ≤ y^2 + z^2 ≤ 1\} $
I have a lot of problems with exercises where I must calculate the Volume of a set using integrals.
Here an example:
Le the set $A$ be $A := \{ (x, y, z) ∈\mathbb{R}^3 : x^2 ≤ y^2 + z^2 ≤ 1\} $. ...
- 900
0
votes
1
answer
128
views
Calculate the part of the sphere that lies inside the cylinder.
Hey I have problems with this problem.
Let $\mathbb{R}^3$ be a sphere, described by $x^2 +y^2 +z^2 ≤ R^2$ , and a cylinder, described by $(x − R/2 )^2 + y^2 ≤ ( R/2 )^2$ .
a) Calculate the part of the ...
- 900
0
votes
1
answer
32
views
show that $g \in L_2((1, \infty))$ defined by $g(x):=\frac{1}{x} \int_{[x,10x]} \frac{f(t)}{t^{\frac{1}{4}}}dλ_1(t).$
I have the following problem:
let $1 < p< 4$ and let $f \in L_p((1, \infty))$.
We consider $g: (1, \infty ) \rightarrow [- \infty, \infty]$ , defined by $g(x):=\frac{1}{x} \int_{[x,10x]} \frac{f(...
- 900
3
votes
1
answer
99
views
Integration by parts including delta function
Often in physics we integrate by parts $$\int_{x_0}^{x_1} f(x) \frac{d}{dx}( \delta(x-y))dx$$
by:
$$=[f(x) \delta(x-y)]_{x_0}^{x_1} - \int_{x_0}^{x_1} \delta(x-y) \frac{df}{dx} dx$$.
I have a really ...
- 727
0
votes
2
answers
48
views
If $g : S \rightarrow \mathbb{R}$ is a continuous function which $\forall s\in S:g(s)\ge 0$ and $\exists s\in S:g(s)>0$, does $\int_{S}g>0$?
I'm trying to prove that if a continuous function $g : S \rightarrow \mathbb{R}$ verifies that $\forall s \in S : g(s) \ge 0$ and $\exists s \in S :g(s) > 0$ where S is an open set that has volume $...
- 430
0
votes
0
answers
43
views
How to define an integral of specific form
Suppose that $\theta(\cdot)$ is a function and it holds that $\frac{\mathrm d}{\mathrm dx}\theta(x)=\theta^{'}(x)$ and
$$\theta^{-}= \max(0,-\theta)$$
in order to define the following integral
$$\int_{...
- 253
1
vote
1
answer
43
views
Using the dominated convergence theorem to show the following equality
I am trying to show the above using the Dominated convergence theorem. I have let the integrand be the limit point of a sequence of functions obtained by replacing the log by its expansion ($\log(\...
- 1,285
4
votes
1
answer
324
views
Dominated convergence theorem for two integrals involving sine
I am stuck on some other problems in introductory measure theory on the convergence theorems (monotone convergence theorem and dominated convergence theorem).
The exercise asks to compute the limit as ...
- 9,762