How to calculate the following integral using spherical coordinates?

The integral goes as follows:

$intintintfrac{z^{2}dxdydz}{x^{2}+y^{2}+z^{2}}$

The domain is limited by the surface $x^2+y^2+z^2=z$

Limit of an integral as a parameter approaches zero

What happens to the following integral as the parameter $varepsilon$ goes to zero (consider a>1)?
$int_{0}^{frac{pi}{4}}frac{e^{-varepsilon e^{itheta}}}{(varepsilon e^{itheta})^{a}}ivarepsilon e^{itheta}dtheta$

integration – Find constants to solve equation with integral

a problem of deriving the right constants to solve an equation.

Find the two constants p and q, so that

$$p∫t^{e-1} ln⁡(e^2 t^{e-1})dt = t^e (q+ln(⁡t) )+C$$

I happen to know the answer, (what should be the answer, anyway), but I can’t find a way of getting there.

$$p=e/(e-1)$$
$$q=(2p-1)/e=(e+1)/(e(e-1))$$
These two are supposed to solve the problem. However, I can’t seem to figure out a good way of deducing this. (Working backwards is not a valid solution.)
Taking the derivative of the right-hand side, and simplifying, this problem reduces to

$$eq+1+e ln⁡(t)=p(2+(e-1) ln(⁡t ))$$

Obviously, some combination of p and q makes this work (exploiting the natural logs somehow?), but how should one proceed to unravel these equations to to find the simple solutions:

$$p=e/(e-1)$$
and
$$q=(e+1)/(e(e-1))$$

pr.probability – The integral of a Gaussian process on a unit sphere

Suppose there exist a zero-mean Gaussian process $mathbb{G} f_u$ indexed by $u in mathcal{S}^{p – 1}$ with known covariance $mathrm{E} big[ mathbb{G} f_u mathbb{G} f_v big]$ when both $u$ and $v$ are known, where $mathcal{S}^{p – 1}$ is the $p$-dimensional unit sphere. Now I want to know what exactly the integral
begin{equation*}
int_{mathcal{S}^{p – 1}} , mathbb{G} f_u , du
end{equation*}

is. This is a integral Gaussian process on the unit sphere. I try my best to find some articles about it, but I cannot find any useful information about it.

Does anyone can help me with how to handle this integral or know some literature about this integral? Thanks so much!

calculus – three dimensional gaussian integral with “non-splittable” term

I need to analytically perform the following integral:

$$mathcal{I} = int_{-infty}^{+infty} e^{frac{-r^2}{2}}(x + y + z)^2$$

I know how to perform multidimensional gaussian integrals by splitting them up and integrating each dimension separately, but the second term is throwing me off. I tried converting to spherical coordinates but to no avail.

calculus – Why this integral give the $F$

I am studying about hydraulic jump in trapezoidal channel:
enter image description here

In the picture where I draw braces I want to find the force to the small triangle. I know the force is product of pressure and Area ($F=PA$) here pressure to the triangle is $P=rho gy$ and in the picture we can see it says $delta F=rho gy. x(Y-y) delta y$
And then integrated it.
I don’t understand why the area of triangle calculated like that. And why the integral with bounds $y=0$ to $Y$ give us the $F$.
I really try to understand the formula instead of memorizing it. So I appriciate any help

ra.rings and algebras – Show that $Z^2 + Y^3 + X^5$ is irreducible in $mathbb C[X,Y,Z].$ Conclude that $B$ is an integral domain

Here is the question I want to answer:

Let $mathbb C(X,Y,Z) cong mathbb C^{(3)}.$ Define rings $$ A = mathbb C(Y,Z)/(Z^2 + Y^3) text{ and } B = mathbb C(X,Y,Z)/(Z^2 + Y^3 + X^5) = mathbb C (x,y,z)$$
where $x,y,z$ are the images of $X,Y,Z$ under the standard projection $mathbb C (X,Y,Z) rightarrow B.$

$(b)$ Show that $Z^2 + Y^3 + X^5$ is irreducible in $mathbb C(X,Y,Z).$ Conclude that $B$ is an integral domain.

Here is my trial:

enter image description here
enter image description here
1- Can anyone give me a feedback on my trial please? specifically,I feel like my reasoning that it is an integral domain is not correct.

2- I got a hint to solve it by finding an isomorphism and noting that the composition of some projections are the same as can be seen below:

(!(enter image description here)(3))(3)

Could anyone show me how to find this isomorphism and its kernel and image please?

EDIT:

3-I also found this question on MSE:

https://math.stackexchange.com/questions/3028756/show-that-fx-y-z-x2-y2z-is-irreducible-in-mathbbcx-y-z?rq=1 but still I am confused about the general procedure and the specific details I should calculate to solve those kinds of problems, could anyone clarify this to me please?

EDIT: I felt like the question is not easy his is why I posted it here.

Uniform estimation of an integral involving a Hölder-continuous function

Let $Omegasubsetmathbb{R}^n$ be open and bounded, let $sin(0,1)$, let $uin C^{0,2s+epsilon}(Omega)$ bounded with $uin C^{0,s}(mathbb{R}^n)$ and such that: $u=0$, on $mathbb{R}^nsetminusOmega$, is true that there exist a constant $C>0$ such that:
$$int_{mathbb{R}^n}frac{|u(x)-u(y)|}{|x-y|^{n+2s}},dyleq C,qquadforall xinOmega,$$
with $C$ that not depend by $xinOmega$. Here $epsilon>0$ is such that $2s+epsilonin(0,1)$, and for every $alpha>0$, $C^{0,alpha}(A)$ is the space of Holder continuous functions on $Asubsetmathbb{R}^n$. Under what assumptions about u is my claim true? I have no idea on how to proceed, any help would be appreciated.

Uniform estimation of an integral

Let $Omegasubsetmathbb{R}^n$ be open and bounded, let $sin(0,1)$, let $uin C^{0,2s+epsilon}(Omega)$ bounded and such that: $u=0$, on $mathbb{R}^nsetminusOmega$, is true that there exist a constant $C>0$ such that:
$$int_{mathbb{R}^n}frac{|u(x)-u(y)|}{|x-y|^{n+2s}},dyleq C,qquadforall xinOmega,$$
with $C$ that not depend by $xinOmega$. I have no idea on how to proceed, any help would be appreciated.

What is the evaluated integral here? [closed]

enter image description here

$$R_n=frac{3pi}nsum_{i=1}^n left(2pi+3pifrac inright)sinleft(2pi+3pifrac inright)$$

HW Question