All Questions
113
questions
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votes
1
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49
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Deal with discontinuity in double integrals
Let $f(x,y)= \frac{x^2+y^2}{x^2}$. Consider
$$ \int_D f(x,y)$$
with $D=\{ (x,y): 0\leq y\leq x, x^2+y^2\leq 1 \}$
The function is unbounded in $D$, due to the denominator. I checked that cannot be ...
- 499
2
votes
2
answers
156
views
Calculate $I = \int_{0}^{\infty} \int_{0}^{\infty} \frac{\ln(xy)}{(x^2 + x + 1)(y^2 + y + 1)} \,dx \,dy$
Question Evaluate $$I = \int_{0}^{\infty} \int_{0}^{\infty} \frac{\ln(xy)}{(x^2 + x + 1)(y^2 + y + 1)} \,dx \,dy$$
My try
$$I = \int_{0}^{\infty} \int_{0}^{\infty} \frac{\ln(x)}{(x^2 + x + 1)(y^2 + y ...
0
votes
0
answers
34
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Is it correct to transform $ \lim_{r\to+\infty}\int_{B(0,r)}\frac{1}{1+x^2+|y|}dx\,dy$ into $\int_{\mathbb{R}^2}\frac{1}{1+x^2+|y|}dx\,dy$?
I just wanted to ask if the provided solution to the following integral is correct.
Prove that
$$
\lim_{r\to+\infty}\int_{B(0,r)}\frac{1}{1+x^2+|y|}dx\,dy=\infty
$$
Solution
The above integral ...
- 547
2
votes
2
answers
110
views
Does the double integral diverge? $\iint_{\Omega} \sqrt{\frac{x^2 + y^2}{xy(1 - x^2 - y^2)}} ~ dx dy$
Does the double integral converge $$\iint_{\Omega} \sqrt{\frac{x^2 + y^2}{xy(1 - x^2 - y^2)}} ~ dx dy$$
I tried using polar coordinate but then half way I realised it was not a good idea so what do I ...
- 1,237
3
votes
1
answer
94
views
How to change coordinates and variables when doing double integral?
Let D be the bounded region in the first quadrant of $R^2$ bounded by curves
$x^2 + 16y^2 = 16, x^2 + 16y^2 = 1, x = y$ and the positive y−axis.
Describe $D$ in the $(u, v)$−plane when $u = x^2 + 16y^...
- 1,237
1
vote
2
answers
68
views
How to prove that $\sup|\int_{(0,1)}\ \cos(tx)dt| < \frac{1}{1+n} $?
Given the function: $$ f(x)=\frac{\sin(x)}x,\qquad\qquad x\in\mathbb R^*$$ and $f(0)=1$.
I have shown that$$f(x) = \int_{(0,1)}\ \cos(tx)dt $$
and that $$f∈ C^{\infty}(\mathbb R).$$
The next question ...
- 61
0
votes
1
answer
44
views
how to prove this change of order of summation and integration
the step I got stuck is:
\begin{equation} \int\limits_0^{\infty}\sum_{y>x}\mathcal{p_\mathit{X}}(y)\,dx=\sum_{y>0}(\int\limits_0^y \mathcal{p_\mathit{X}}(y)\,dx) \end{equation}
the context is, $...
- 77
0
votes
0
answers
51
views
How to simplify or evaluate an integral inside an integral in R?
I have this double integral to work with:
$$
\int_a^b f(x) \int_0^x g(t)(x-t)dtdx
$$
I am rather very weak in calculus and not sure how should I approach to simplify or evaluate it.
For numerical ...
- 119
3
votes
1
answer
106
views
Calculate double integral $\iint_D\frac{\sqrt{x^2+y^2 - a^2}x}{(x^2+y^2)^2}dxdy$ over unbounded region $D$
Calculate
$$I = \iint\limits_D\frac{\sqrt{x^2+y^2 - a^2}x}{(x^2+y^2)^2}\,dx\,dy$$
where the region of integration is:
$$D = \{(x,y) \in \mathbb{R}^2 \mid x+y - a\sqrt{2} \geq 0, -x+y +a\sqrt{2} \geq 0,...
- 89
0
votes
0
answers
40
views
Estimate an integral with a logarithmic weight
Let $\epsilon \in (0,1)$ and $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have
$$
\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\...
- 297
0
votes
0
answers
57
views
differentiate under integral sign says something went wrong
$$I=\int_0^\infty \frac{\tan^{-1} ax \tan^{-1}bx}{x^2}\mathrm dx$$
$$\frac{dI}{da}=\int_0^{\infty} \frac{\tan^{-1} bx}{x^2} \cdot \frac{x}{(ax)^2+1}\mathrm dx$$
$$\frac{d^2I}{\mathrm da \mathrm db} =\...
user953847
2
votes
0
answers
53
views
Integral estimate in $\mathbb{R}^N$
Let $0<s<1<p<\infty$ and define
$$
L^{p-1}_{ps}(\mathbb{R}^N):=\left\{v\in L^{p-1}_{\mathrm{loc}}(\mathbb{R}^N):\int_{\mathbb{R}^N}\frac{|v(y)|^{p-1}}{1+|y|^{N+ps}}\,dy<+\infty\right\}.
...
- 431
1
vote
1
answer
57
views
Improper multivariable integral calculation
Given a domain
$$D=\{(x,y)\mid 0≤x≤5,y≤x≤3y\}$$
how can I calculate the integral $$\bbox[5px,border:2px solid #C0A000]{\iint_D \frac y x \, dx\, dy}$$
using a $D_n$ domain series?
I have tried to ...
- 29
0
votes
1
answer
75
views
convergence test for improper integrals
in an old book on calculus in the chapter about improper integrals there was this theorem that says when $\int_{a}^{\infty }f(x)dx $ can be written in the form
$\int_{a}^{\infty} \frac{\phi (x)}{x^k}...
- 21
0
votes
0
answers
97
views
Integrating the Dirac delta of a polar rose $\delta(r-\sin(2\theta))$
I'm trying to integrate the Dirac delta of the polar rose $r(\theta)=\sin(2\theta)$, i.e. $$\delta(r-\sin(2\theta)).$$
If my reasoning isn't wrong right from the beginning, this should give me the ...
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