A sin^2 and cos^2 integrals question

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In summary, the conversation is about the definite integral over the squared sine function and its average value. The first question asks if the integral is equal to half of the upper limit, and the second question suggests that the function's positivity makes the integration limits irrelevant. However, the responses state that the integral does not equal half of the upper limit and that the average value of the sin/cos functions is not always 1/2. The conversation then moves on to discussing the range and solution to the equation for the cosine function. The final question asks if the statement that the integral over squared sine/cosine is always 1/2 is correct and for what range, to which the response is no and that the range can be determined by solving
  • #1
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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  • #2
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.
 
  • #3
are you trying to help?

I'm asking because I don't know the answers.
 
  • #4
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...
 
  • #5
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?
 
  • #6
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.
 
  • #7
thanks!
 

Related to A sin^2 and cos^2 integrals question

1. What is the difference between sin^2 and cos^2 in the context of integrals?

In an integral, sin^2 and cos^2 refer to the squared values of the sine and cosine functions, respectively. While sin^2 represents the square of the sine function, cos^2 represents the square of the cosine function.

2. How do I solve an integral involving sin^2 or cos^2?

To solve an integral involving sin^2 or cos^2, you can use trigonometric identities such as the double angle formula or the power reducing formula. These identities allow you to rewrite the integral in terms of simpler trigonometric functions, making it easier to solve.

3. Can I use a calculator to evaluate sin^2 and cos^2 integrals?

Yes, most scientific calculators have built-in functions for evaluating sin^2 and cos^2 integrals. However, it is important to know the underlying principles and techniques used to solve these integrals, as relying solely on a calculator may not always be accurate or efficient.

4. Are there any special cases when solving sin^2 and cos^2 integrals?

Yes, there are a few special cases that may arise when solving sin^2 and cos^2 integrals, such as when the limits of integration are not in the standard range of 0 to 2π or when the integrand contains other trigonometric functions. In such cases, you may need to use additional trigonometric identities or make substitutions to simplify the integral.

5. How can I apply sin^2 and cos^2 integrals in real-world problems?

Sin^2 and cos^2 integrals have various applications in physics, engineering, and other fields. For example, they can be used to calculate the area under a curve, the volume of a solid, or the work done by a force. Additionally, they can be used to model periodic phenomena such as sound waves or electrical signals.

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