Do the Lebesgue-Stieltjes integral and the Riemann integral have the same rules about the change of order of integration? I mean I know how to deal with Riemann integral, but I’m not sure if I can simply apply the same rules to the Lebesgue-Stieltjes. Thanks. It’s double integration in $\mathbb{R}^2$, by the way.

Question: Let $f:[a.b]\to \Bbb R \in R[a,b]$ s.t. $f(x)>0 \ \forall x \in \Bbb R.$ Is $\int _a^b f(x)\,dx>0$ ? What We thought: We know how to prove it for weak inequality, for strong inequality – no clue ðŸ™‚

This question is closely related to one I just asked here. I believe that it is just different enough to warrant another question; please let me know if it does not. In the question mentioned above, I was informed by Joriki that $$\int \cos\left(\frac{1}{x}\right) \mathrm{d}x = x \cos\left(\frac{1}{x}\right) + \operatorname{Si}\left(\frac{1}{x}\right)$$ where $$\mbox{Si}(u) = \int \frac{\sin(u)}{u} […]

Can one explain why taking an integral of $\frac 1{|y|}$ over a ball in $\mathbb{R}^3$ of radius $r$ is equal to a constant times $r^2$? If $y \in \mathbb{R}^3$, then $$\int_{B(0,r)} \frac{dy}{|y|}=Cr^2$$ where $C$ is an appropriately chosen constant. Why is this true? (This is part of a proof in Â§12.3.2 of PDE Evans, 2nd […]

Let $u\in W^{1,2}_0((a,b))$ and $\psi’$ the derivative of a convex function $\Psi\in C^1(\mathbb{R})$. If I want to consider the antiderivative of $\psi'(u)$, what happens with the $u$ inside $\Psi(u)$? Is it possible to say how does the antiderivative looks like? I need this for an other proof. Regards

I am given the following integral: $\int_C {e^{(-1+i)\log(z)}}$ with $C:|z|=1$ $\operatorname{Log} (z): 0\le \operatorname{arg} (z)\le 2\pi$ Is it possible to resolve this integral using Cauchy theorem? The function is not analytic in a line inside C so my guess would be that it is not possible. Thanks for the help

Here is Theorem 6.19 (change of variable) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $\varphi$ is a strictly increasing continuous function that maps an interval $[ A, B]$ onto $[ a, b]$. Suppose $\alpha$ is monotonically increasing on $[ a, b]$ and $f \in \mathscr{R}(\alpha)$ on $[a, b]$. Define […]

Please help me with this integral: $$\int{\frac{x^2 + 4}{x^2 + 6x +10}}\,dx .$$ I know I must solve it by substitution, but I don’t know how exactly.

If i have frustum and its top is cut by an inclined plane at angle $\alpha$, such that it makes an ellipse. The height is $h$ (at the axis of obliquely truncated frustum). How can i use triple integral to determine its volume, and coordinates of its geometric center. I will be thankful. I can […]

Let $\star$ denote convolution binary operation and $\otimes$ denote cross correlation binary operation between two functions. Let $f,g,h$ be functions. Does this negative distribution property holds? $$f\star ( g\otimes h) = – (f \star g) \otimes h$$ Edit : It is given that $f$ is an odd function. Definitions : $$f\star g = \int \limits_{-\infty}^\infty […]

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