# Tag Info

## New answers tagged multiple-integral

### 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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Accepted

• 139k
1 vote

### 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 ...
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1 vote

$$I:=\iint_{D} {x^2-y^2 \over 1+x^4+y^4 } \mathrm{d}x \mathrm{d}y$$ 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 $$-I=\iint_{... • 10.4k 1 vote Accepted ### Flux of a vector field through a closed surface Your solid T can be imagined by rotating the region in the xz-plane defined by$$\{(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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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}...