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Multiple integral involving exponential function

To motivate ourselves, consider the following problem: select $k$ Gaussian random variables $X_{1},\cdots,X_{k}$, such that $P(X)=\frac{1}{\sqrt{\pi}}e^{-X^{2}}$, what is the probability that they are ...
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Find the domain to make the integral max

Yes, the maximum value is attained for the domain $C:=\{(x,y): x^2+y^2\leq 1\}$, i.e. the disk centered at the origin of radius $1$. Indeed, if $D$ is any measurable set in $\mathbb{R}^2$ then $$\...
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How to find the limit of $a_n=\int_{\mathcal{D}}(3xy-x^2y-xy^2)^n\lambda_2(dx dy)$ on $\mathcal{D}=\{(x,y)\in\mathbb{R}^2|x\gt0,y\gt0,x+y\lt3\}$

As a corollary of the beta integral formula, you have for nonnegative integers $a$ and $b$ that $$\int_0^1 x^a (1-x)^b \,dx = \frac{a! b!}{(a + b + 1)!}.$$ Now as a further corollary of this, let us ...
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How to find the limit of $a_n=\int_{\mathcal{D}}(3xy-x^2y-xy^2)^n\lambda_2(dx dy)$ on $\mathcal{D}=\{(x,y)\in\mathbb{R}^2|x\gt0,y\gt0,x+y\lt3\}$

Note that $$\begin{align} a_n&=\iint_{\mathcal{D}}(xy)^n (3-x - y)^n dx dy\\ &=\iint_{\mathcal{D}}(xy)^n\left(\int_{z=0}^{3-x-y}nz^{n-1}dz\right) dxdy\\ &=n\int_{x=0}^3\int_{y=0}^{3-x}\...
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Rigorous proof that $dx dy=r\ dr\ d\theta$

Along with other commenters, I was also confused by the accepted answer of @geodude. I'd never heard of wedge or exterior products before. But now I understand and can explain why @geodude's answer ...
  • 249
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Evaluation of $~\iint_{D} {x^2-y^2 \over 1+x^4+y^4 } \mathrm{d}x \mathrm{d}y~$ where $~D~$is of bounded and closed and line symmetric with $~y=x$

$$ I:=\iint_{D} {x^2-y^2 \over 1+x^4+y^4 } \mathrm{d}x \mathrm{d}y $$ Swap the names $x$ and $y$ then $$ I=\iint_{D} {y^2-x^2 \over 1+x^4+y^4 } \mathrm{d}y \mathrm{d}x $$ Then notice that $$ -I=\iint_{...
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Flux of a vector field through a closed surface

Your solid $T$ can be imagined by rotating the region in the $xz-$plane defined by $$\{(x,0,z):x\leq z \leq \sqrt{x},0\leq x \leq 1\}$$ about the $z-\text{axis}$. In cylindrical coordinates this can ...
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Double Integral with a Delta Function

As a slightly more general case, let us evaluate the integral for the region $g_1(x)<y<g_2(x)$ and $x_1<x<x_2$. Thus our integral takes the form \begin{equation} \int_{x_1}^{x_2}\mathrm{d}...

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