0
votes
0answers
17 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
71 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
3answers
97 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
42 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
74 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
246 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
56 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
74 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
63 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
29 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
134 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
56 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
124 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
121 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
110 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
226 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
82 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
601 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
953 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 ...