Why Is Int|4sin(x) + sin(10x)| dx > Int|4sin(x)|dx ?
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From: Bret Cahill on 6 Aug 2010 11:20 > >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 ">=." 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? 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: Frederick Williams on 6 Aug 2010 13:05 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 ">=." For all x? -- I can't go on, I'll go on.
From: mjc on 6 Aug 2010 13:36 On Aug 6, 10:05 am, Frederick Williams <frederick.willia...(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 ">=." > > 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.
From: Bret Cahill on 6 Aug 2010 14:16 > > > > >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. The likely reason behind the inequality is that the high frequency lobes don't cancel out near the low frequency crossings. Instead the function |4sin(x) + sin(10x)| is positive and fairly large when |4sin(x) is near zero. A second attempt on the sometimes erratic Wolfram Alpha to eliminate the lobes by just integrating over the 0.3 to (pi - 0.3) seemed to make Int|4sin(x) + sin(10x)|dx very close to Int|4sin(x)|dx -- not a formal math proof but it helps confirm the large-positive-lobes-near- the-crossings graphical explanation. Bret Cahill
From: Oppt on 6 Aug 2010 15:51
On Fri, 6 Aug 2010 11:16:30 -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 ">=." >> >> > 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. 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)? |