2
votes
0answers
45 views

Proving a set is of measure zero.

Let $C\subset A\times B$ be a set of content zero. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B: (x,y)\in C\}$ is not of content zero. Show that $A'$ is a set of measure zero. ...
1
vote
1answer
22 views

How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ? I know absolutely continuous means ...
1
vote
1answer
90 views

Show there exists $x\in (0,1)$ such that $f(x) \leq \int_0^1 f(t) dt$

Please help me check my proof, thanks! (a) Show there exists $x\in (0,1)$ such that $$f(x) \leq \int_0^1 f(t) dt.$$ Proof: when $f$ is constant a.e, the equality holds for all points except for a ...
1
vote
1answer
71 views

Proof about Riemann integrability of a bounded function

I tried to prove the following, please could somebody tell me if my proof is correct? If $f: [a,b]\to \mathbb R$ is a bounded Riemann integrable function then for every $\varepsilon > 0$ there ...
2
votes
0answers
53 views

Contour integral: different answers with different contours

Good day to everyone. I have a following contour integral problem. I have to find a solution for the integral $$\underset{\gamma_r }{\oint }\frac{e^{\lambda s} }{(1-s) s^{a-b} \left(s-\theta ...
1
vote
1answer
41 views

Requirements for integration by parts/ Divergence theorem

In order to use the integration by parts formula(or more generally the divergence theorem) for functions of several variables $$\int_{\Omega} \nabla u\cdot v d \Omega = \int_{\partial \Omega}(u(v ...
0
votes
1answer
27 views

Unique solution for $\int_x^1 f(t) dt = 2x$ and $|x| < \epsilon$

Let $f$ be continuous on $\mathbb{R}$ such that $$f(0) \neq -2 \quad\text{ and } \quad \int_0^1 f(t) = 0.$$ Show that there exists $\epsilon > 0$ such that the equation $$\int_x^1 f(t) dt = 2x$$ ...
2
votes
0answers
52 views

$f\in L^1(0,\infty)$ monotone, show $\lim_{x\rightarrow \infty} xf(x) = 0$ [duplicate]

Here is the solution: First $f$ is monotone and integrable on $(0,\infty)$, wolg we can assume that $f>0$ and approaches $0$ as $x$ goes to infinity. Observe that $$xf(2x) \leq \int_x^{2x} ...
0
votes
2answers
34 views

Vanishing moments and integrability

Is this correct? $\int_\mathbb{R}x^m f(x) dx=0 \iff \int_\mathbb{R}x^m \overline{f(x)}\,dx =0$. If yes then please tell the conditions under which this holds.
1
vote
1answer
11 views

Clarification about equality regarding integrals

I'm reading Brezis ch. 8 and got stuck in a passage of lemma 8.2 pag. 205. Let $I=(a,b)$, let $g \in L_{loc}^1(I)$, for a fixed $y_0 \in I$, set $$ v(x) = \int_{y_0}^x g(t)dt \ \ \ \ \ \ \ \ \ x \in ...
4
votes
3answers
182 views

Integration by change the variable

Let, $\int_{-1}^1\sqrt{1+e^x}\operatorname{dx}$. Write as an integral of a rational function and compute it. Suggest: change the variable in order to eliminate the square root. My work was: ...
1
vote
0answers
32 views

If $f(x)$ is integrable on $[a,b]$ then $c\cdot f(x)$ is also integrable and $\int_a^b c\cdot f(x) dx=c\cdot \int_a^b f(x) dx$

I proved the first part of this theorem which says that $c\cdot f(x)$ is integrable,but how to prove that $\int_a^b c\cdot f(x) dx=c\cdot \int_a^b f(x) dx$? Maybe it provides a bit help if i tell how ...
4
votes
1answer
71 views

Proof verification: $\int_a^x f(t) \text{dt}=0$, $f$ is continuous at $x$. Prove that $f(x)=0$

Let $f:[a,b]\to R$ be an integrable function such that for all $x \in[a,b]$, we have $\int_a^x f(t) \text{dt}=0$. Show that if $f$ is continuous at $x \in [a,b]$, then $f(x)=0$. My attempt: argue ...
0
votes
1answer
33 views

Would like to compute the limit of some integral

I was working on a exercise where the goal was to compute the following limit, $$\lim_{n\rightarrow\infty}\int_{\mathbb R}e^{-|x|n}e^{-\frac{x^2}{2}}dx$$ and some tutor of mine claimed that the limit ...
0
votes
0answers
34 views

Area of solid of known cross-section

I was looking at surface areas of solids of known cross-section (that is, you take a region in the xy plane, set up cross sections perpendicular to the region that are defined as a function of x, and ...
3
votes
3answers
58 views

Find dimension of the intriguing vector space

We are given vector space of polynomials over $\mathbb R$ of two variables with powers not higher than 2013. Let's consider subspace $V$ which contains such polynomials $f$, so following holds for ...
0
votes
1answer
42 views

Proving $f(x) = f(0) + f'(0)x + \int_0^x (x-t) f''(t) dt$ for all x

Suppose f has a continuous second derivative. Prove $f(x) = f(0) + f'(0)x + \int_0^x (x-t) f''(t) dt$ for all x. Can someone check this for me? What I started with is that if we let $h(x) = \int_0^x ...
1
vote
0answers
25 views

Check my answer - show a function is integrable and find the integral

Let $Q =[0,1]$x$[0,1]$. Let $f: Q \to \mathbb R$ defined as such: if $(x,y) \in \mathbb Q$x$\mathbb Q$ then $f(x,y)=\frac{1}{n_1}+\frac{1}{n_2}$ where $x=\frac{m_1}{n_1}$ and $y=\frac{m_2}{n_2}$ are ...
1
vote
1answer
47 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
1
vote
0answers
27 views

Under which assumptions we have $f\in L^p$ for all $p\in\mathbb N$

So here is my question, I wanted to generalize, under what assumptions for some $f$ we have $f\in L^p(\mathbb R)\;\forall p\in\mathbb N.$ And I found the following, Let $f\in L^p(\mathbb R)$ for ...
2
votes
1answer
113 views

Question about integration on a box

Let $Q \subseteq \mathbb{R}^n$, and $f: Q \to \mathbb{R} $ is integrable over $Q$. $f \geq 0$. if $A \subseteq Q$, then $\int_Q f \geq \int_A f $ Attempt: say $\epsilon > 0$ Let $P_1$ be a ...
1
vote
0answers
24 views

Continuity of a function on a box $Q$ implies integrability

Let $Q \subseteq \mathbb{R}^n$ be a box, and say $f: Q \to \mathbb{R}$ is continuous, then $f$ is integrable on $Q$. MY ATTEMPT: Since $f$ is continuous on $Q$, then $f$ must be uniformly continuous ...
0
votes
1answer
51 views

Prove the integral of $\cot x$ is $-\ln|\csc x|+C$

I know how to prove it to be $\ln|\sin x|+C$, but I do not know the method to prove it this way. thanks
1
vote
1answer
53 views

how do you show one definite integral is less than another definite integral over the same region?

How do you show $$ \int ^{b}_{a} f(x) \,dx \leq \int ^{b}_{a} g(x) \,dx$$ if you know f(x) and g(x) are continuous over [a,b] and $$f(x) \leq g(x)$$ for $${a \leq x \leq b}$$ Here is the way I ...
0
votes
0answers
53 views

$f \in \mathcal{R}(\alpha)$ on $[a,b]$, then $\exists P_n$ s.t. $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$.

Assume $f \in \mathcal{R}(\alpha)$ on $[a,b]$, and prove that there are polynomial $P_n$ such that $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$. This is what I have, ...
0
votes
2answers
487 views

Prove that $\int_0^{2\pi} \cos \theta e^{\cos\theta} \cos(\sin \theta) - \sin \theta e^{\cos\theta} \sin(\sin \theta)\,\mathrm{d}\theta$ equals zero.

Note this question is a part of a bigger question, I have done a) and b). Problem 2 Let $u(x,y) = x e^x \cos y - y e^x \sin y $ a) Show that u is harmonic in the entire plane b) Find a ...
1
vote
1answer
33 views

Domination $\Rightarrow$ $0$ equality

Let $\phi \in C^{\infty}_c(\mathbb{R})$. In class, my teacher said that the dominated convergence theorem (DOM) may be used to prove that $$ \lim_{\epsilon \to 0^+} \int_{-\epsilon}^{\epsilon} \! \log ...
1
vote
1answer
122 views

Seeking domination?

Define $f_n : \mathbb{R} \to \mathbb{R}$ by $$ f_n(x) := \int_{-n}^{nx} \! \frac{\sin t}{t} \, \mathrm{d}t. $$ I'm searching for a function $g \in L^1_{\text{loc}}(\mathbb{R})$ (that is, a real ...
5
votes
2answers
77 views

$f$ integrable $\Leftrightarrow f<\infty$ a.s.?

$f\colon\Omega\to\mathbb{R}$ measurable function on measure space$(\Omega,\mathfrak{A},\mu)$. I am interested to know if then $$ f\text{ is integrable }\Leftrightarrow f\text{ is finite a.s.}~~~. $$ ...
3
votes
2answers
95 views

Proof of integral

Is there an analytical method to show that $$ \int_{-a}^a\exp\left(\frac{-1}{1-(x/a)^2}\right)\,\mathrm{d}x=ka, $$ for $a>0$. I have confirmed this result numerically for a range of values of $a$. ...
3
votes
1answer
37 views

Integrability of time differences via bootstraping?

the question is somehow inspired by the Alt-Luckhaus paper (Lemma 1.5) Let $B:\mathbb{R}\to\mathbb{R}$ be continuos and nonnegative, $\Omega\subset \mathbb{R}^n$ a bounded domain, $h,T>0$. Let ...
2
votes
1answer
45 views

Error in theorem 3-8 “Calculus on manifolds”

This error is also commented in the Addenda of the book, with a proposed solution, but I don't understand how it should fix the proof. The (part of the proof of the) theorem involved is: Suppose ...
1
vote
0answers
44 views

Non-negative, continuous function with integral [duplicate]

Let there be an integrable, non-negative function $f$ in a range $[a,b]$. If the integral $\int_a^b f(x) \, dx$ equals $0$, prove that $f(x)=0$ for every $x$ for which $f$ is continuous. I have ...
0
votes
0answers
69 views

Box-sum criterion related problem

Define a function $f$ by $$f(x) = \begin{cases}42 & \text{if }x =1,2,3,4; \\0 & \text{otherwise} \end{cases}$$ Prove that $f$ is integrable on $[0,5]$ by using the box-sum criterion. My ...
2
votes
0answers
65 views

Proving the converse of the Cauchy criterion for integration

Prove the converse of the Cauchy criterion for integration. That is, prove that if $f$ is integrable on $[a,b]$, then for any $\epsilon>0$ there is a $\delta>0$ so that for any partitions ...
1
vote
0answers
108 views

$\int f^2(x)dx=0$ if $\int f(x) x^n=0 $ for all $n=1,2,3,\dots$ (TIFR GS ($2012$))

Question is to check if for a continuous function $f: [0,1]\rightarrow \mathbb{R}$ $$\int_0^1 f^2(x)dx=0 \text { given that }\int_0^1 f(x) x^n=0 ~\forall ~n=0,1,2,3,\dots$$ I think it is true and ...
0
votes
2answers
83 views

Show that $\int_{-x}^xf=2\int_0^xf$ implies that $f$ is even.

Please tell me whether the following proof works: $f$ is continuous on $\mathbb R\to\mathbb R$ such that $\int_{-x}^xf=2\int_0^xf~\forall~x\in\mathbb R.$ Show that $f$ is an even function. ...
1
vote
1answer
117 views

Show that the fundamental solution $E_2$ of Laplace equation is local integrable

For $x\in\mathbb{R}^n\setminus\left\{0\right\}$ the function $$ E_n(x):=\begin{cases}\frac{1}{2\pi}\ln\left(\frac{1}{\lVert x\rVert}\right), & \text{ for ...
1
vote
0answers
95 views

Let $f:[a,b]\to\mathbb R$ be bounded and for all $c\in(a,b),~f$ is Riemann integrable on $[c,b].$ Prove that $f$ is Riemann integrable on $[a,b].$

Please tell me whether my proof for the following problem will work: Let $f:[a,b]\to\mathbb R$ be bounded and for all $c\in(a,b),~f$ is Riemann integrable on $[c,b].$ Prove that $f$ is Riemann ...
8
votes
1answer
192 views

Question on Riemann sums

Question is : What I have read in Riemann sum definition is something like $$\sum_{i=1}^n f(y) .|x_i-x_{i-1}|$$ So, at first sight i am afraid this is not even related to Riemann integration of ...
3
votes
2answers
91 views

What am I doing wrong?

I am trying to prove the integral test for series, but got a strange result. Assume that $f$ is decreasing and positive. Because the series can be imagined as the area-sum of $1$-wide rectangles of ...
2
votes
1answer
56 views

Proof that the Riemann-Integral satisfies $\int_A \lambda f = \lambda \int_A f$

Suppose $A\subset\mathbb{R}^n$ is a closed rectangle and $f:A\to \mathbb{R}$ is Riemann-Integrable on $A$. I want to show that $\lambda f$ is integrable and that $$\int_A \lambda f =\lambda\int_Af $$ ...
3
votes
4answers
138 views

Show that $\int_{1}^\infty \dotsb\int_{1}^\infty \frac{dx_1 \dotsb dx_n}{x_1^{\alpha_1}+\dotsb + x_n^{\alpha_n}}<\infty$

Here's my solution to an old qualifier problem. Would you tackle it differently? Is there a flaw in my work? Suppose that $\alpha_1, \dotsc, \alpha_n$ are positive numbers such that ...
0
votes
0answers
56 views

Is this integral a counter example to this theorem?

I may have misunderstood the proposition, but I thought it was: Let $f$ be a function $[a,b]\times I\to \Bbb R$, where $I$ is some real interval. Then a sufficient condition for ...
3
votes
1answer
119 views

$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx=-\frac{1}{\cos (\varphi )^2}$ is that correct?

Good day. This integral looks very simple, yet I don't know how to start. $$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx$$ I know that if the lower integration limit was $-\infty$ it would ...
3
votes
2answers
99 views

Prove the Fundamental Theorem of Calculus

Prove the Fundamental Theorem of Calculus with this hypothesis: If $f$ is integrable over $[a,b]$, if $g:[a,b]\rightarrow\Bbb R$ given by $g(x)=\int_{a}^{x}f(t)dt$ and $f$ is continuous in $x_0 \in ...
0
votes
1answer
49 views

change of variables while integrating

Suppose I have an integral that looks like: $$I=\int_{r=0}^\infty\int_{\omega_1=-\infty}^\infty\int_{\omega_2=-\infty}^\infty ...
4
votes
2answers
77 views

Integration of Rational Functions - Problem with proof relating to complex solutions

$\quad$I was reviewing integration of rational functions, and all was going well until I saw this bit, in the end of the explanation: (translated from portuguese by me) $ \qquad \qquad\text{with ...
2
votes
2answers
87 views
4
votes
2answers
97 views

Calculate the volume of $T = \{(x,y,z) \in \mathbb R^3 : 0 \leq z \leq x^2 + y^2, (x-1)^2 + y^2 \leq 1, y \geq 0\}$

Calculate the volume of $T = \{(x,y,z) \in \mathbb R^3 : 0 \leq z \leq x^2 + y^2, (x-1)^2 + y^2 \leq 1, y \geq 0\}$ so I said that the integral we need is $\iint_{D} {x^2 + y^2 dxdy}$. But when I ...