A sin^2 and cos^2 integrals question

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SUMMARY

The discussion centers on the evaluation of definite integrals for the functions sin²(x) and cos²(x). The claim that the integral of sin²(x) from 0 to a equals a/2 is incorrect; the average value of these functions is not always 1/2. The participants clarify that the integration range significantly impacts the results, and specific formulas must be applied for accurate calculations. The conversation highlights the importance of understanding integral calculus fundamentals when addressing such problems.

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  • Integral calculus fundamentals
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  • Knowledge of definite integrals
  • Familiarity with average value of functions
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  • Study the properties of definite integrals for trigonometric functions
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  • Explore specific integral formulas for sin²(x) and cos²(x)
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dudinka
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hi!

1. Is the definite integral over [sin(x)]^2 from 0 to a, is a/2? and also the cosine?

2. Is it because the average of the sin/cos functions is always 1/2?

3. and if so, the integration region doesn't matter because the function is positive?
I mean, what about integration from -a to a?

4. and from -3a to 0.5a? is it: 0.5*[3.5a]?

thanks!
 
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Well, for 1. why do you ask us ? Can't you compute it all by yourself ? The answer is "no", btw. Then you can figure out the aswers to the other questions. And the average of the sin/cos in not always 1/2.
 
are you trying to help?

I'm asking because I don't know the answers.
 
Well, the simple calculation of those 2 integrals doesn't yield the results you proposed. So do you know anything about integral calculus ? I'm asking because the answer will make me post the solutions to those integrals or not...
 
I've just found a formula on the net for those integrals, thnks anyway.

but I remember my teacher said that integral over squared(!) sin/cos is always 1/2 or something.
Is this correct? if so, for what range?
 
Nope, of course not. The range can be let's say for the cos^2 from 0 to the solution of the equation 2a+sin 2a=2.
 
thanks!
 

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