3
votes
1answer
45 views

Show the integrals are alike without explicit computation.

I saw a couple of striking integrals which are $\hspace{5em} \displaystyle \color{black}{\displaystyle \int_a^b \frac{x\,\mathrm{d}x}{\sqrt{(x^2-a^2)(b^2-x^2)}} }$$\displaystyle\ =\ $$\displaystyle ...
0
votes
1answer
26 views

Showing that the composite trapezoid rule is invariant to certain symmetry

I am trying to show that if $f$ holds certain symmetric properties than the composite trapezoid rule $$ T(n) = (b-a)\frac{f(a)+f(b)}{2n} + \frac{b-a}{2} \sum_{k=1}^{n-1} f(x_k) $$ is invariant over ...
0
votes
1answer
65 views

why are these integrals equal?

I've been trying to analyse Steiner symmetrization. I've asked before about the symmetrization preserving $\textit{volume}$, I'm still going over the same proof. I think I understand a bit what is ...
15
votes
2answers
171 views

slick way of transforming an integral?

The function $$ (\alpha,\beta) \mapsto \int_0^\beta \frac{\sin\alpha\,d\zeta}{1+\cos\alpha\cos\zeta} $$ is a symmetric function of $\alpha$ and $\beta$. But I don't know a simpler way to see that ...
4
votes
3answers
591 views

Why isn't an odd improper integral equal to zero

My calculus book says that the integral of $\frac1x$ cannot cross zero. Now it seems obvious that because of symmetry, there will always be an interval whose integrals are equal in magnitude and ...
1
vote
0answers
67 views

Dilation Invariance

Given the formula $$F(x)= \sum_{n=-\infty}^{\infty}f(x+n) $$ We know that is invariant under translations of the form $y=x+n$ for any integer $n$. However can we find a similar formula for dilations ...