Articles of integration

Definitions of multiple Riemann integrals and boundedness

Given a function $f:D\subset\mathbb{R}^n\to\mathbb{R}$ let $\bar{f}:R\to\mathbb{R}$, where $R$ is a hyperrectangle such that $D\subset R$, be defined by $$\bar{f}(\boldsymbol{x}) = \begin{cases} f(\boldsymbol{x}), & \boldsymbol{x}\in D \\ 0, & \boldsymbol{x}\in R\setminus D \end{cases}$$ If there is a number $I\in\mathbb{R}$ such that, for any partition $P$ (whose mesh is $\delta_P$) of $R$ into hyperrectangles $R_i$ of volume […]

function whose limit does not exist but the integral of that function equals 1

Is there a function whose limit does not exist as x approaches infinity but the integral of that function from negative infinity to positive infinity is equal to 1?

Proof for transformation of random variables formula

Suppose we have $U=g_1(X,Y)$, $V=g_2(X,Y)$ for two random variables, X,Y. I want to prove this formula: $$f_{U,V}(u,v)=f_{X,Y}(h_1(u,v), h_2(u,v))|J|$$ Question 1 Is this right? $$f_{U,V} = \frac{\partial}{\partial u \partial v} P(U < u, V < v)$$ by the FTC. $$= \frac{\partial}{\partial u \partial v} \int_{\{(x,y) : \\ g_1(x,y) < u,\\ g_2(x,y) < v \}} f_{X,Y}(x,y)dxdy$$ $$= […]

Calculating volume enclosed using triple integral

Calculate the volume enclosed between $x^2 + y^2=1$, $y^2+z^2=1$, $x^2+z^2=1$ I am supposed to solve this question using multiple integral. I am not able to visualize the resultant figure of whose I am supposed to calculate the volume.

Mollifiers: Derivative

This thread is meant as lemma for: Semigroups & Generators: Entire Elements: Construction Given a smooth mollifier: $\varphi\in\mathcal{L}(\mathbb{R}): \varphi’\in\mathcal{L}(\mathbb{R})$ Do the derivatives exist in the sense: $$f\in\mathcal{C}(\mathbb{R}):\quad\int_{-\infty}^{\infty}\frac{1}{h}\left\{\varphi(\hat{x}+h)-\varphi(\hat{x})\right\}f(\tfrac{1}{n}\hat{x})\mathrm{d}\hat{x}\to\int_{-\infty}^{\infty}\varphi'(\hat{x})f(\tfrac{1}{n}\hat{x})\mathrm{d}\hat{x}\quad(\|f\|_\infty<\infty)$$ (Note the unbounded domain of integration.)

The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$

Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be $$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$ Evaluating such integrals, especially for small $n$, is essentially a straightforward exercise in complex analysis (integrate on a semicircle in the UHP using the residue theorem, send the radius to $+\infty$ and show that the integral on […]

How to integrate this type of fractions

Q: How to integrate this type of fractions $$\int^{\frac{\pi}{2}}_{\frac{\pi}{3}} \frac{\sin\frac{\theta}{2}}{1+\sin\frac{\theta}{2}} d\theta $$ What should I do here? I don’t even know where to start from. Please help me by giving me a hint.

Mean of gamma distribution.

So I was trying to prove the mean result of gamma distribution which is $\frac{\alpha}{\lambda}$. My attempt, $E(X)=\int_{0}^{\infty }x f(x)dx$ $=\int_{0}^{\infty } \frac{\lambda^{\alpha}}{\Gamma (\alpha)}x^{\alpha}e^{-\lambda x}dx$ After integrating it, I got the result $$\frac{\lambda^{\alpha}}{\Gamma (\alpha)} \cdot\frac{\alpha}{\lambda}(\int_{0}^{\infty } x^{\alpha-1}e^{-\lambda x}dx)$$. I’m stuck here. Could anyone continue it for me and explain? Thanks a lot.

How to evaluate$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy$, where $D$ is the sphere in 3D?

I would like to evaluate this integral over the surface of a sphere in 3D: $$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy. $$ It seems there is a lot of symmetry in this integral so I imagine there is a good chance there is an explicit solution. However, I usually deal with 2D Helmholtz problems so […]

$\int { (x^{3m}+x^{2m}+x^{m})(2x^{2m}+3x^{m}+6)^{1/m}}dx$ if $x>0$

For any natural number $m$,how to evaluate $\int { (x^{3m}+x^{2m}+x^{m})(2x^{2m}+3x^{m}+6)^{1/m}}dx$ if $x>0$ ?