1
vote
1answer
37 views

Improper integral $\int_{B}\frac {1}{|x|^\alpha}dV$

Let B be the ball $|x|\le 1$, $x\in R^n$. For what $\alpha$ does $$\int_{B}\frac {1}{|x|^\alpha}dV$$ exists? I find it hard when it comes to generalize this statement in $R^n$. I've been able to do ...
4
votes
0answers
52 views

Improper multivariable integrals

I'm having trouble with the integral $$\iiint_{1\le x^2+y^2+z^2 }\frac{\mathrm{d}x~\mathrm{d}y~\mathrm{d}z}{xyz}$$ this is what I've done so far: $$\lim_{b\to +\infty}\int_1^b \int_0^{2\pi} ...
1
vote
1answer
69 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
4
votes
2answers
87 views

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$ I thought of evaluating the iterated integral ...
2
votes
1answer
48 views

Improper double integral evaluation by changing the order of integration

I was watching an MIT OCW recitation video about exchanging the order of integration on double integrals. The example was: ...
3
votes
1answer
38 views

Show $\iint xye^{-xy}\,dx\,dy$ is convergent or divergent

Determine convergence/divergence of $$\iint xye^{-xy}\,dx\,dy$$ for $x,y \geqslant 0$ i.e. in the first quadrant. I have managed to show that $xye^{-xy} \to 0$ in the first quadrant but other ...
4
votes
1answer
78 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
1
vote
0answers
44 views

Integration in polar coordinates?

Given $$ A=\begin{pmatrix} a & b \\b & c \end{pmatrix}, x=(x_1,x_2), (Ax,x)>0 $$ and $$(x,y)=x_1\cdot y_1+x_2\cdot y_2$$ I'm trying to prove that $$ \int_{-\infty}^\infty ...
1
vote
1answer
29 views

Explain what the teacher did, convergence of improper integral

I'd like someone to explain what the teacher did, because I'm not sure I understand. Basically, the question is for which values of $p$, does the integral $$\int_{1}^{\infty} \frac{dt}{t \log ...
4
votes
4answers
85 views

$\iint_{\mathbb R^2} \frac{dx \, dy}{1+x^{10}y^{10}}$ diverges or converges?

Question I'm trying to solve to prepare for an exam. I need to find out if $\displaystyle\iint_{\mathbb R^2} \frac{dx\,dy}{1+x^{10}y^{10}}$ diverges or converges. What I did: I switched to polar ...
3
votes
1answer
70 views

Check my proof $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sin(x^2+y^2) \, dx \, dy$ diverges

I am trying to prove that $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sin(x^2+y^2)\, dx\,dy$ diverges and I did it like this: $x=r\cos \theta$, $y=r\sin\theta$, $\theta \in [0,2\pi]$, $r\in ...
2
votes
0answers
36 views

Difficult integrals, do they converge, show there's no dependence on parameters.

I am trying to figure out whether these integrals: a) $$\int_{\mathbb R^2}{{\rm d}\xi \over \left\vert\vphantom{\Large A}\,\log\left(\left\vert\,x - \xi\,\right\vert\right) -\log\left(\left\vert\,y ...
1
vote
3answers
105 views

Evaluate $\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\,dx\,dy$

I would like to compute the following, $$ \int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy $$ It is obvious that we can rewrite the integral above to, $$ ...
4
votes
3answers
176 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 ...
2
votes
3answers
96 views

How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy$ be found?

How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy $ be found, if $D$ is $x$ O $y$ axis? So far I have done it this far: ...
0
votes
0answers
123 views

Taking the improper integral of a series term-by-term.

Suppose I have a function $f$ which appears in some complicated formula in a term that looks like $\int_{-\infty}^{\infty} f(x, v)\,\text{d}v$. Basically, I want to write $f$ in a series of functions ...
6
votes
2answers
110 views

$\int_{0}^{\infty}\int_{1}^{\infty}\frac{x^2-y^2}{(x^2+y^2)^2}dxdy$ diverges?

I want someone to review my proof that $$\int_{1}^{\infty}\int_{0}^{\infty}\frac{x^2-y^2}{(x^2+y^2)^2}dxdy$$ does not converge. To make things easier, I said let's look at the entire first quadrant ...
4
votes
1answer
79 views

Double integral involving zeta function: $\int_0^\infty \frac{1-12y^2}{(1+4y^2)^3}\int_{1/2}^{\infty}\log|\zeta(x+iy)|~dx ~dy.$

I'm having trouble evaluating the following double integral: $$\int_0^\infty \frac{1-12y^2}{(1+4y^2)^3}\int_{1/2}^{\infty}\log|\zeta(x+iy)|~dx ~dy.$$ Do please remark that $\zeta$ is the zeta ...
2
votes
0answers
51 views

Multivariable Integral Calculus help

I have two questions. First: Is my proof "strong" enough? I am being asked to prove that $$\int_{0}^\infty\int_{0}^x e^{-sx}f(x-y,y) dydx = \int_{0}^\infty\int_{0}^\infty e^{-s(u+v)}f(u,v) dudv$$ ...
0
votes
0answers
20 views

Convergent Improper Integral help

I am currently studying improper integrals and came across the following problem. Analyze the convergence of the improper integral of $f(x,y) = 1 / ( x^4 + y ^2 ) $ over $R = \{(x,y) : x\geq 1, y\geq ...
0
votes
0answers
19 views

Divergence of a triple integral

How to prove the following result (if ture, in the first place)? $$\nabla\cdot\int\int\int ...
2
votes
1answer
117 views

Integration limits of the double integral after conversion to the polar coordinates

I want to solve the following double integral: $$\int_0^{\infty}dx\int_{-\infty}^{\infty}dy\,f(x,y).$$ And for example I made a conversion to the polar coordinates, $x=r\cos{\theta}$ and ...
3
votes
4answers
144 views

Evaluation of a particular type of integral involving logs and trigonometric function

Is there any closed form for $$ \int _0 ^{\infty}\int _0 ^{\infty}\int _0 ^{\infty} \log(x)\log(y)\log(z)\cos(x^2+y^2+z^2)dzdydx$$ if yes then how to prove it?
2
votes
1answer
43 views

Is $\iint\limits_{D}\frac{1}{(x^{2}+y^{2})^{5}+5}dx\,dy$ convergent in $D=[0,+\infty)\times[0,+\infty)$

Can you tell me whether my approach is correct. We first switch to polar coordinates and we get the following integral: ...
4
votes
1answer
82 views

Inverse Laplace

Hi how to verify the following I tried substitution and integration by parts but can bot figure it out.. $$\int_0^{\infty} \exp(- \lambda t ) \frac{x}{\sqrt{2\pi t^3}}\exp(-\frac{x^2}{2t}) dt = ...
1
vote
1answer
300 views

Why does Fubini's theorem not apply in this example?

Let $f(x,y)=\frac{x-y}{(x+y)^3}$ and $g(x,y)=\text{sgn}(x-y)e^{-|x-y|}$ ( http://en.wikipedia.org/wiki/Sign_function ) We have $$\int_0^1\int_0^1f(x,y)dy dx = 1/2 = -\int_0^1\int_0^1f(x,y)dy dx$$ ...
2
votes
0answers
79 views

Fubini theorem for improper Riemann integral

Is there a version of Fubini's theorem for improper Riemann integrals? Here's an example of what such a version might look like. If $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is bounded and non-negative ...
2
votes
1answer
87 views

$\int_0^\pi\int_0^\infty e^{-xy}\sin kx~dy~dx=$ ?

$$\int_0^\pi\int_0^\infty e^{-xy}\sin kx~dy~dx=~?$$ My computation is $$\int_0^\infty e^{-xy}\sin kx~dx=\frac{1}{k+y^2}$$ so $$\int_0^\pi\frac{1}{k+y^2}dy=\frac{\sqrt{k}}{k}\arctan\pi$$ ...
1
vote
1answer
68 views

Continuity of a function defined by an integral

Ok, Here's my question: Let $f(x,y)$ be defined and continuous on a $\le x \le b, c \le y\le d$, and $F(x)$ be defined >by the integral $$\int_c^d f(x,y)dy.$$ Prove that $F(x)$ is continuous on ...
2
votes
0answers
31 views

Integral to be computed

I am interested in computing $$J(a,b):=\int_{\mathcal{I}(a,b)}\frac{dx \ dy\ dz}{|ax^3+ay^2+bz^3|^{2/3}}, $$ where $a,b$ are natural numbers and $$\mathcal{I}(a,b):=(0,1]^3\cap\{x,y,z \in ...
2
votes
1answer
154 views

Improper Multiple Integral

I am trying to solve the following exam problem: Let $s$ be a real number. Find the condition under which the improper integral $$I:=\iint_{\mathbb R^2} \frac{dxdy}{(x^2-xy+y + 1)^s}$$ converges, ...
0
votes
2answers
57 views

how to find this type of definite double integral?

could any one tell me how to find this type of definite double integral? $$\int_{0}^{\infty}\int_{x}^{\infty}{e^{{-y\over2}}\over y}dydx$$ Thank you.
4
votes
1answer
142 views

When are the following multiple improper integrals convergent?

This is a question from a past exam. For which $p, q\in \mathbb R$ do the following improper integrals converge? ...
2
votes
1answer
129 views

A parameterized elliptical integral (Legendre Elliptical Integral)

$$ K(a,\theta)=\int_{0}^{\infty}\frac{t^{-a}}{1+2t\cos(\theta)+t^{2}}dt $$ For $$ -1<a<1;$$ $$-\pi<\theta<\pi$$ I know this integral to be a known tabulated Legendre elliptic integral, ...
1
vote
1answer
70 views

Converging/Diverging Integrals

Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "divergent": $$\int_4^\infty \frac{1}{x^2+1}\,dx.$$ I am so lost. I ...
0
votes
1answer
119 views

Question about L2 Inner Product and Integrals

Does exists $f\in L_2(\mathbb{R}^d)$ such that for all $g\in L_2(\mathbb{R}^d)$ which is not identically zero: ...
2
votes
2answers
246 views

Change of variables in improper double integral - how to find the limits?

Consider the following integral (originating from the product of the Laplace transforms of $f$ and $g$): $$\int_0^\infty \int_0^\infty f(u)\ g(v) e^{-s(u+v)}\ du\ dv.$$ For this integral, the ...
0
votes
2answers
85 views

Determine $\alpha >0$ for which $\iint_Af(x,y)^\alpha dx \, dy < +\infty$

Let $A=\{(x,y)\in \mathbb R^2: 0<x<1, 0< y < \sqrt{x}\}$ and $f \colon A \to\mathbb R$ a continuous function s.t. $$ \frac{1}{x^2+y^2} \le f(x,y) \le\frac{2}{x^2+y^2} $$ for every ...
2
votes
1answer
649 views

Iterated Integrals and Unbounded Regions

Context I am having difficulty finding the posterior distribution of a Bayesian model with two parameters, which involves evaluating a double integral over an unbounded region. I prefer not to post ...
5
votes
2answers
973 views

Evaluating a convergent improper triple integral over the unit sphere

In Exercise 5 (f) of Angus Taylor's Advanced calculus (p. 659) one is asked to find the value of the following integral if convergent: $$I:=\underset{R}{\iiint}\dfrac{x^2 y^2 z^2}{r^{17/2}}\mathrm ...