How to Calculate the Average Value of a Sinusoid Over a Given Interval?

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Discussion Overview

The discussion revolves around calculating the average value of a sinusoidal function over a specified interval, specifically from -π/5 to π/5. The conversation includes mathematical reasoning and exploration of properties of sinusoidal functions.

Discussion Character

  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant inquires about the method to calculate the average value of a sinusoid over the interval from -π/5 to π/5.
  • Another participant presents the formula for the average value of a function over an interval, emphasizing the need for the function to be properly integrable.
  • A further contribution explains the relationship between the average value and the area under the curve for both constant and variable functions, linking it to the integral of the function over the interval.
  • In response to the initial inquiry, a participant notes that since the sine function is odd, the average value over a symmetric interval around zero will be zero.

Areas of Agreement / Disagreement

Participants express differing views on the average value of the sinusoid, with one asserting it is zero due to the properties of the sine function, while others focus on the mathematical process without reaching a consensus on the average value itself.

Contextual Notes

The discussion does not resolve the implications of the sine function's properties on the average value calculation, nor does it clarify the assumptions regarding the integrability of the function over the specified interval.

perryben
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If I had a sinusoid, how would I find the average value of it over a given interval. Say -pi/5 to pi/5 for instance. Thanks everybody.
 
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The average value of a function, say [tex]f(x),[/tex] over the interval [a,b] is given by the the formula

[tex]f_{\mbox{ave}}=\frac{1}{b-a}\int_{x=a}^{b}f(x)dx[/tex]

where I have assumed that [tex]f(x),[/tex] is properly integrable over [a,b].
 
The point is: if you had a constant function, f(x)= c, the "area under the curve" from a to b would f(x)(b-a)= c(b-a). With a variable function, that area is [tex]\int_a^b f(x)dx[/tex].

If fave is the average of the function we must have
[tex]\int_a^b f(x)dx= f_{ave}(b-a)[/tex]
 
If I had a sinusoid, how would I find the average value of it over a given interval. Say -pi/5 to pi/5 for instance. Thanks everybody.
The sine function is odd. Therefore the average over an interval symmetric around 0 will be 0.
 

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