Articles of integration

Changing the order of integration (for Lebesgue-Stieltjes integral and Riemann integral)

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.

Integral of a positive function is positive?

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 🙂

How does $\int_1 ^x \cos(2\pi/t) dt$ have complex values for real values of $x$?

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} […]

Integration of $|y|^{-2}$ over the ball $B(0,r)$

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 §12.3.2 of PDE Evans, 2nd […]

antiderivative of $\psi'(u)$ for $u\in W^{1,2}_0((a,b))$

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

Cauchy Integral Theorem and the complex logarithm function

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

Theorem 6.19 in Baby Rudin: Do we need the continuity of $\varphi$?

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 […]

Solve integral $\int{\frac{x^2 + 4}{x^2 + 6x +10}dx}$

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.

Volume of frustum cut by an inclined plane at distance h

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 […]

Does negative distributive property of convolution over cross correlation holds?

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 […]