I would like to find the (four-dimensional) volume of the region given by $$xy>zw \quad\wedge \quad x>-y \quad\wedge \quad x^2+y^2+z^2+w^2<1,$$ for $x,y,z,w\in\mathbb{R}$ and where the last condition means that the whole thing is bounded by the unit $4$-ball. The boundary of the region given by the two first conditions would be (I think) \begin{align} xy=zw […]

In page 122 of a book by William J. LeVeque, namely Topics in Number Theory (1956), there is an exercise for evaluating the following integral in two ways. $$\int_0^1\!\!\!\int_0^1\frac1{1-xy}\,dy\,dx$$ First way is to write the integrand as a geometric series, $$\int_0^1\!\!\!\int_0^1\frac1{1-xy}\,dy\,dx=\int_0^1\!\!\!\int_0^1\left(\sum_{n=1}^\infty(xy)^{n-1}\right)\,dy\,dx=\sum_{n=1}^\infty\frac1{n^2}$$ and the second way by use of a suitable change of variables ($y:=u-v,x:=u+v$) which […]

Problem: Find the centroid of the part of the large loop of the limacon $r=1+2\cos(\theta)$ that does not include the small loop. I know that in order to compute the centroid one needs to use following equations $$x=\frac{\iint_D xdydx}{m}$$ and $$y=\frac{\iint_D ydydx}{m},$$ where $$m=\iint_D dydx.$$ Note that we are assuming $\rho=1,$ which should work for […]

I want to calculate the asymptotic form as $x \to 0$ of the following integral. \begin{alignat}{2} I_2(x) &=&& \int_0^{\infty}du\int_0^{\infty}dv\, \frac{1}{(u+v)^2}\exp\left(-\frac{x}{u+v}\right) \\ &=&& \frac{\partial^2}{\partial x^2} \int_0^{\infty}du\int_0^{\infty}dv\, \exp\left(-\frac{x}{u+v}\right) \end{alignat} How can we solve? This question is related with this post. Thanks.

I’m having a hard time with this problem, i have some ideas, but i don’t know how to continue. Here is the exercise: “Consider the solid $S$ bounded by the walls of the superior cone whose equation is $z = \sqrt{3}\sqrt{x^2+y^2} $ and inside the sphere of equation $x^2+y^2+(z-2)^2=4$ whose density at a point $P$ […]

Please, help me to prove that: $$ \int_Q \int \frac{dxdy}{x^{-1} + |\ln y| – 1} \leq 1,$$ where $$ Q = [0; 1] \times [0; 1] $$ Any ideas how to start. Thank you.

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