Why Is Int|4sin(x) + sin(10x)| dx > Int|4sin(x)|dx ?
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From: Bret Cahill on 6 Aug 2010 23:41 > >> > > > >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. > > >> > > It seems to be true, at least if the ">" is replaced with ">=." > > >> > For all x? > > >> > -- > >> > I can't go on, I'll go on. > > >> If you consider an interval on which sin(x) and sin(10x) have > >> different signs (such as [pi/10, pi/10 + pi/20]), the inequality is > >> false. > > >The absolute value keeps everything positive so the inequality should > >hold over long intervals or over one or more complete [low frequency] > >cycles if the amplitude of the high frequency term is significant. > > [snip] > > Is the question whether or not the inequality is true for *any* > interval from x=a to x=b (b > a)? If so, then as the post by mjc > points out, the inequality is false. It may be possible to find some interval where it isn't true, especially for the more general case of high amplitude/low frequency + low amp/high freq. The piso is kicking in so I'm pretty neutral on the debate right now. > Do you have in mind that > an interval have some x=a and x=b that are constrained, say for > example that a = n*pi (an integer multiple of pi)? Interval selection is possible, it just might not be necessary. Bret Cahill
From: David C. Ullrich on 7 Aug 2010 08:58 On Fri, 6 Aug 2010 08:20:34 -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. > >It seems to be true, at least if the ">" is replaced with ">=." It can't be true because it's meaningless! Both integrals are only _defined_ up to addition of an arbitrary constant; you can't say that f(x) + c_1 >= g(x) + c_2 without knowing what c_1 and c_2 are. And it's clearly not true for Int_a^b for arbitrary a and b. It may well be true for Int_0^{2 pi} >Increase the high frequency term's amplitude a little and the >difference between the two functions increases a lot. > >> Where does the question come from? > >Same as any math proof, i.e., why does a^2 + b^2 = c^2? Huh? I asked where you got the question from. See, the way you put it it's false. So the question is whether the source stated it incorrectly or you just didn't quote the source correctly. Whether it was stated that way or not, I guarantee you that the original author _intended_ to be talking about Int_0^{2 pi}. >A formal proof isn't necessary here, just some graphical explanation. >At first glance it seems that if 4sin(x) + sin(10x) is "centered" on 4 >sin(x) then the + and - contributions from the sin(10x) would pretty >much cancel out. > >Even eliminating the multiple crossings from high frequency component >from the interval doesn't seem to change the situation much. > >What's even more interesting is the paucity of hits you get on google >with just a few key words like math + identity + integral + trig + >absolute + value. > >It may be hard to sell ad space using math identities as a draw. > > >Bret Cahill > > >
From: Bret Cahill on 7 Aug 2010 11:37 > >> >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. > > >It seems to be true, at least if the ">" is replaced with ">=." > > It can't be true because it's meaningless! Both integrals are > only _defined_ up to addition of an arbitrary constant; You were supposed to assume they are definite integrals over the the same x interval. .. . . > >Increase the high frequency term's amplitude a little and the > >difference between the two functions increases a lot. > > >> Where does the question come from? > > >Same as any math proof, i.e., why does a^2 + b^2 = c^2? > > Huh? I asked where you got the question from. No way! You're not getting any account numbers either. Bret Cahill
From: Frederick Williams on 12 Aug 2010 18:23 Bret Cahill 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. > > > > >It seems to be true, at least if the ">" is replaced with ">=." > > > > It can't be true because it's meaningless! Both integrals are > > only _defined_ up to addition of an arbitrary constant; > > You were supposed to assume they are definite integrals over the the > same x interval. Were we also supposed to assume that the interval was... um... what? -- I can't go on, I'll go on.
From: Virgil on 12 Aug 2010 18:58
In article <4C647454.AF398046(a)tesco.net>, Frederick Williams <frederick.williams2(a)tesco.net> wrote: > Bret Cahill 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. > > > > > > >It seems to be true, at least if the ">" is replaced with ">=." > > > > > > It can't be true because it's meaningless! Both integrals are > > > only _defined_ up to addition of an arbitrary constant; > > > > You were supposed to assume they are definite integrals over the the > > same x interval. > > Were we also supposed to assume that the interval was... um... what? Note that there ARE intervals on which 4*sin(x) and sin(10*x) are of opposite sign. |