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Math Help - Squared sin and cos definite integrals question

  1. #1
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    Question Squared sin and cos definite integrals question

    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. for example, from -3a to 0.5a? is it: 0.5*[3.5a]?

    thanks!
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  2. #2
    MHF Contributor red_dog's Avatar
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    \displaystyle\int_0^a\sin^2xdx=\int_0^a\frac{1-\cos 2x}{2}dx
    Can you take it from here?
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  3. #3
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    thanks for that!

    but the thing with Int[sin^2] is 1/2 or something... what am I missing?
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  4. #4
    MHF Contributor red_dog's Avatar
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    \displaystyle\int_0^a\sin^2xdx=\int_0^a\frac{1-\cos 2x}{2}dx=\left.\frac{x}{2}\right|_0^a-\left.\frac{\sin 2x}{4}\right|_0^a=\frac{a}{2}-\frac{\sin 2a}{4}
    So, the result depends on a.
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