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Math Help - Convert discrete formula to continuous formula

  1. #1
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    Convert discrete formula to continuous formula

    By using approximation is it enough proof to show equation left and right is related to each other?

    Also, what are the method usually be used to convert equation on right to left?
    I have done the question but i would like to see other method as well.
    Attached Thumbnails Attached Thumbnails Convert discrete formula to continuous formula-approximate-pv1.jpg  
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  2. #2
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    Re: Convert discrete formula to continuous formula

    Hey chuackl.

    What do you mean by related exactly? Do you mean that they are approximately the same?
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  3. #3
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    Re: Convert discrete formula to continuous formula

    Yes, if the change is small, both are approximately the same.
    However, I would interested to know the way to convert the equation on the right to the left by mean of integration.
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  4. #4
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    Re: Convert discrete formula to continuous formula

    You need to get something in the form of a summation of f(x)dx or f(x)d(g(x)). If g(x) is continuous, then you get a Riemann Integral but if it is more general (discontinuous for example or not differentiable), then you need to look at the Lebesgue Integral or other similar integrals.

    Integrals are just summations involving limits, so once you can get your function into the above form (summing rectangles as the width goes to zero), then you can start to look at the integral form.

    To do this you should look at taking your function and expanding it as some kind of Taylor series or equivalent where you get the right summation structure to turn it into an integral.
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  5. #5
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    Re: Convert discrete formula to continuous formula

    Thank you for showing the other way of integration.


    Please refer attachment in "integration 2" i found that te original equation(linear equation) can be convert from this way to another form of equation.
    Just need to substitute P3=P1+dp and V=dv solve for the rest of the equation.

    This give me an idea also a discreate equation in 2nd attachment can convert into continuous equation by using the same way as in 1st attachment.

    Please verify is this way of doing still obey mathematical rule?
    Attached Thumbnails Attached Thumbnails Convert discrete formula to continuous formula-pv-gamma.jpg   Convert discrete formula to continuous formula-integration-2.jpg  
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  6. #6
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    Re: Convert discrete formula to continuous formula

    sorry for pushing.

    Is the every approach correct and make sense?
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