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Math Help - absolute trig integral

  1. #1
    Member akhayoon's Avatar
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    absolute trig integral

    Hi, I'm getting really weird answers for the answer of this integral

    \int_{0}^{\pi}|\frac{1}{2}+cosx|~dx
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by akhayoon View Post
    Hi, I'm getting really weird answers for the answer of this integral

    \int_{0}^{\pi}|\frac{1}{2}+cosx|~dx
    \int_{0}^{\pi} \left | \frac{1}{2}+cosx \right | ~dx = \int_0^{2\pi/3}\left ( \frac{1}{2} + cos(x) \right )~dx - \int_{2\pi/3}^{\pi} \left ( \frac{1}{2} + cos(x) \right )~dx

    I get \sqrt{3} + \frac{\pi}{6}.

    -Dan
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by akhayoon View Post
    Hi, I'm getting really weird answers for the answer of this integral

    \int_{0}^{\pi}|\frac{1}{2}+cosx|~dx
    remember what absolute values mean. |x| = \left \{ \begin{array}{cc} x & \mbox{ if } x \ge 0 \\ & \\ -x & \mbox{ if } x < 0 \end{array} \right.

    thus we need to find where our function (without absolute values) is negative, and negate its integral on that region. we can find that 1/2 + cos(x) is negative for [2pi/3, pi], thus:

    \int_0^{\pi} \left| \frac 12 + \cos x \right|~dx = \int_0^{\frac {2 \pi}3} \left( \frac 12 + \cos x \right)~dx - \int_{\frac {2 \pi}3}^{\pi} \left( \frac 12 + \cos x \right)~dx
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  4. #4
    Member akhayoon's Avatar
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    oh so does the whole 1/2+cosx change to -1/2-cosx?

    or does just cosx change to -cosx
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by akhayoon View Post
    oh so does the whole 1/2+cosx change to -1/2-cosx?

    or does just cosx change to -cosx
    notice that we have 1/2 + cos(x) in brackets with a minus sign in front of it. what does that mean?
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  6. #6
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    Hello, akhayoon!

    Did you make a sketch?


    \int_{0}^{\pi}\left|\frac{1}{2}+\cos x\right|\,dx

    It is the cosine curve raised a half-unit.
    Code:
            |
            *
            |:::*
            |:::::*
            |::::::*
            |:::::::
            |:::::::*
            |::::::::       π
         ---+--------o------+---
            |        ↑*::::::
            |        ↑  *::::
            |       2π/3    *
            |

    I totally agree with Dan's integrals and his final answer.

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