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Math Help - integration

  1. #1
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    Smile integration

    Hi there

    I was wondering if anyone would be able to help me with a problem I'm having. I've nearly finished a sum and I know the answer but I just haven't managed to quite get there. Any help would really be appreciated.

     \int_{0}^{2\pi} ( \frac{9}{8} - \frac{1}{8} \cos(4\theta)) \: d\theta \: = \: \int_{0}^{2\pi} ( \frac{9}{8}) \: d\theta - \int_{0}^{2\pi} (\frac{1}{8} \cos(4\theta)) \: d\theta

    The answer  \frac{189 \sqrt{2} \pi}{8} but as far as I can figure out \: \int_{0}^{2\pi} ( \frac{9}{8}) \: d\theta becomes ( \frac{9}{8}) \: \theta but does  \int_{0}^{2\pi}(\frac{1}{8} \cos(4\theta)) \: d\theta become (\frac{1}{8})(\frac{1}{4} \sin(4\theta) when I integrate as I've been trying to get to the answer I've been given but can't as isn't sin(4\theta)=0 when \theta = 2\pi ??

    If anyone could help me with the last step, I'd really appreciate it

    Thanks
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  2. #2
    Super Member wingless's Avatar
    Joined
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    Quote Originally Posted by michaela-donnelly View Post
    Hi there

    I was wondering if anyone would be able to help me with a problem I'm having. I've nearly finished a sum and I know the answer but I just haven't managed to quite get there. Any help would really be appreciated.

     \int_{0}^{2\pi} ( \frac{9}{8} - \frac{1}{8} \cos(4\theta)) \: d\theta \: = \: \int_{0}^{2\pi} ( \frac{9}{8}) \: d\theta - \int_{0}^{2\pi} (\frac{1}{8} \cos(4\theta)) \: d\theta

    The answer  \frac{189 \sqrt{2} \pi}{8} but as far as I can figure out \: \int_{0}^{2\pi} ( \frac{9}{8}) \: d\theta becomes ( \frac{9}{8}) \: \theta but does  \int_{0}^{2\pi}(\frac{1}{8} \cos(4\theta)) \: d\theta become (\frac{1}{8})(\frac{1}{4} \sin(4\theta) when I integrate as I've been trying to get to the answer I've been given but can't as isn't sin(4\theta)=0 when \theta = 2\pi ??

    If anyone could help me with the last step, I'd really appreciate it

    Thanks

    Your calculation looks good to me. And after all,  \int_{0}^{2\pi}(\frac{1}{8} \cos(4\theta)) \: d\theta = 0. So the answer is only \int _0^{2\pi }\frac{9}{8}\theta~ \text{d}\theta = \boxed{~\frac{9}{4}\pi~}

    Is this whole question? I think it isn't, the answer you gave is not the result I found.
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