Why Is Int|4sin(x) + sin(10x)| dx > Int|4sin(x)|dx ?
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From: Bret Cahill on 5 Aug 2010 19:25 Is there a list of identities on integrals of absolute values of sums of trig functions somewhere? Bret Cahill
From: Bret Cahill on 6 Aug 2010 01:01 Graphically it seems the +/- "perturbations" from the high frequency component would cancel and in fact, smaller amplitudes of the higher frequency component rapidly cause the integral to approach Int|4sin(x)| dx. The extra "lobes" from crossing the axis might explain it. > Is there a list of identities on integrals of absolute values of sums > of trig functions somewhere? Bret Cahill
From: Oppt on 6 Aug 2010 04:58 On Thu, 5 Aug 2010 16:25:04 -0700 (PDT), Bret Cahill <BretCahill(a)peoplepc.com> wrote: >Is there a list of identities on integrals of absolute values of sums >of trig functions somewhere? > > >Bret Cahill You asked about Int |4sin(x) + sin(10x)| dx > Int |4sin(x)| dx I'm assuming those vertical bars mean 'absolute value'. I think the inequality should be >= (greater than or equal to). Evaluate each of the integrals f(x) = Int [0..x] 4 sin(u) + sin(10 u) du g(x) = Int [0..x] 4 sin(u) du and compare f(x) and g(x) on the interval from x=0 to x=pi. Is it true that f >= g on that interval? Do other intervals need to be considered? Is it true that 1/5 sin(5x)^2 >= 0 for all real x? Is that interesting?
From: Bret Cahill on 6 Aug 2010 10:39 > >Is there a list of identities on integrals of absolute values of sums > >of trig functions somewhere? > > >Bret Cahill > > You asked about Int |4sin(x) + sin(10x)| dx > Int |4sin(x)| dx > > I'm assuming those vertical bars mean 'absolute value'. Yes. > I think the inequality should be >= (greater than or equal to). Yes. > Evaluate each of the integrals > > f(x) = Int [0..x] 4 sin(u) + sin(10 u) du > g(x) = Int [0..x] 4 sin(u) du > and compare f(x) and g(x) on the interval from x=0 to x=pi. It looks like the "lobes" from the small amplitude high frequency component of f(x) occupy as much area above g(x) as below g(x) and therefore would cancel out over a pi length interval. > Is it true that f >= g on that interval? Do other intervals need to > be considered? If the difference between f(x) and g(x) was caused by the multiple crossings caused by the small amplitude high frequency component then grapihically it seems it might be easier to approximate g(x) with f(x) by omitting the multiple crossings region from the interval. Select the interval to be less than pi and centered at pi/2 or 3 pi/2 or 5 pi/ 2, etc. This helps very little, however, in trying to get g(x) from f(x). > Is it true that 1/5 sin(5x)^2 >= 0 for all real x? Yes. Bret Cahill
From: David C. Ullrich on 6 Aug 2010 10:47
On Thu, 5 Aug 2010 16:25:04 -0700 (PDT), Bret Cahill <BretCahill(a)peoplepc.com> wrote: >Is there a list of identities on integrals of absolute values of sums >of trig functions somewhere? Saying Int|4sin(x) + sin(10x)| dx > Int|4sin(x)|dx is meaningless. Where does the question come from? In particular, is it actually a statement about Int_0^{2 pi} ? > >Bret Cahill > > > > |