# Tagged Questions

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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$\int_x^1 f(t) dt = 2x$$ ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $h(x) = \int_0^x ... 0answers 25 views ### Check my answer - show a function is integrable and find the integral Let$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}$are ... 1answer 47 views ### Let$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}$be a continous function. We'll now define a sequence ... 0answers 27 views ### Under which assumptions we have$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)$for ... 1answer 113 views ### Question about integration on a box Let$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$be a ... 0answers 24 views ### Continuity of a function on a box$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$must be uniformly continuous ... 1answer 51 views ### Prove the integral of$\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 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 ... 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, ... 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 $a) Show that u is harmonic in the entire plane b) Find a ... 1answer 33 views ### Domination$\Rightarrow0$equality Let$\phi \in C^{\infty}_c(\mathbb{R})$. In class, my teacher said that the dominated convergence theorem (DOM) may be used to prove that $$\lim_{\epsilon \to 0^+} \int_{-\epsilon}^{\epsilon} \! \log ... 1answer 122 views ### Seeking domination? Define f_n : \mathbb{R} \to \mathbb{R} by$$ f_n(x) := \int_{-n}^{nx} \! \frac{\sin t}{t} \, \mathrm{d}t. $$I'm searching for a function g \in L^1_{\text{loc}}(\mathbb{R}) (that is, a real ... 2answers 77 views ### f integrable \Leftrightarrow f<\infty a.s.? f\colon\Omega\to\mathbb{R} measurable function on measure space(\Omega,\mathfrak{A},\mu). I am interested to know if then$$ f\text{ is integrable }\Leftrightarrow f\text{ is finite a.s.}~~~. $$... 2answers 95 views ### Proof of integral Is there an analytical method to show that$$ \int_{-a}^a\exp\left(\frac{-1}{1-(x/a)^2}\right)\,\mathrm{d}x=ka, $$for a>0. I have confirmed this result numerically for a range of values of a. ... 1answer 37 views ### Integrability of time differences via bootstraping? the question is somehow inspired by the Alt-Luckhaus paper (Lemma 1.5) Let B:\mathbb{R}\to\mathbb{R} be continuos and nonnegative, \Omega\subset \mathbb{R}^n a bounded domain, h,T>0. Let ... 1answer 45 views ### Error in theorem 3-8 “Calculus on manifolds” This error is also commented in the Addenda of the book, with a proposed solution, but I don't understand how it should fix the proof. The (part of the proof of the) theorem involved is: Suppose ... 0answers 44 views ### Non-negative, continuous function with integral [duplicate] Let there be an integrable, non-negative function f in a range [a,b]. If the integral \int_a^b f(x) \, dx equals 0, prove that f(x)=0 for every x for which f is continuous. I have ... 0answers 69 views ### Box-sum criterion related problem Define a function f by$$f(x) = \begin{cases}42 & \text{if }x =1,2,3,4; \\0 & \text{otherwise} \end{cases}$$Prove that f is integrable on [0,5] by using the box-sum criterion. My ... 0answers 65 views ### Proving the converse of the Cauchy criterion for integration Prove the converse of the Cauchy criterion for integration. That is, prove that if f is integrable on [a,b], then for any \epsilon>0 there is a \delta>0 so that for any partitions ... 0answers 108 views ### \int f^2(x)dx=0 if \int f(x) x^n=0 for all n=1,2,3,\dots (TIFR GS (2012)) Question is to check if for a continuous function f: [0,1]\rightarrow \mathbb{R}$$\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$$I think it is true and ... 2answers 83 views ### Show that \int_{-x}^xf=2\int_0^xf implies that f is even. Please tell me whether the following proof works: f is continuous on \mathbb R\to\mathbb R such that \int_{-x}^xf=2\int_0^xf~\forall~x\in\mathbb R. Show that f is an even function. ... 1answer 117 views ### Show that the fundamental solution E_2 of Laplace equation is local integrable For x\in\mathbb{R}^n\setminus\left\{0\right\} the function$$ E_n(x):=\begin{cases}\frac{1}{2\pi}\ln\left(\frac{1}{\lVert x\rVert}\right), & \text{ for ... 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 ... 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 ... 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 ... 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 $$\int_A \lambda f =\lambda\int_Af$$ ... 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 ... 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 ... 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 ... 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 ...
Suppose I have an integral that looks like: I=\int_{r=0}^\infty\int_{\omega_1=-\infty}^\infty\int_{\omega_2=-\infty}^\infty ...