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Math Help - Definition of Integral

  1. #1
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    Definition of Integral

    Could somebody please explain how, from the definition of integral, we can deduce the following:

     \displaystyle \int_a^{a+h} f(x) \, dx = f(a)h + \frac{f'(a)}{2} h^2 + \frac{f''(\xi)}{6}h^3

    Obviously some form of the Taylor expansion with Lagrangian Remainders is going on, but where does the multiplying by h on the RHS come into it, why are the numerators moved 'back' one term than they usually are in a Taylor expansion, and how do we arrive at this result 'from the definition of integral'?

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  2. #2
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    Quote Originally Posted by AnonymitySquared View Post
    Could somebody please explain how, from the definition of integral, we can deduce the following:

     \displaystyle \int_a^{a+h} f(x) \, dx = f(a)h + \frac{f'(a)}{2} h^2 + \frac{f''(\xi)}{6}h^3

    Obviously some form of the Taylor expansion with Lagrangian Remainders is going on, but where does the multiplying by h on the RHS come into it, why are the numerators moved 'back' one term than they usually are in a Taylor expansion, and how do we arrive at this result 'from the definition of integral'?

    AnonymitySquared
    Substitute f(x) = f(a) + f'(a) (x - a) + \frac{f''(a)}{2} (x - a)^2 +  \frac{f^{(3)} (a)}{6} (x - a)^3 + .... and integrate term-by-term.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Substitute f(x) = f(a) + f'(a) (x - a) + \frac{f''(a)}{2} (x - a)^2 +  \frac{f^{(3)} (a)}{6} (x - a)^3 + .... and integrate term-by-term.
    Actually, I got it, in a manner that is a bit less laborious than that!

     \int_{a}^{a+h} f(x)\,dx = g(a+h)-g(a)

     g(a+h) = g(a) + g'(a)h + \frac{g''(a)h^2}{2} + \frac{f''(\xi)h^3}{6}

     \xi \in [x_0,x]

     \therefore g(a+h) - g(a) = g'(a)h + \frac{g''(a)h^2}{2} + \frac{g'''(\xi)h^3}{6}

     \therefore \int_{a}^{a+h} f(x)\,dx =  g'(a)h + \frac{g''(a)h^2}{2} + \frac{g'''(\xi)h^3}{6}

    By the fundamental theorem of calculus:

     g'(a) = f(a) \, \, \, g''(a) = f'(a) \, \, \, g'''(a) = f''(a) , hence:

    \int_{a}^{a+h} f(x)\,dx =  f(a)h + \frac{f'(a)h^2}{2} + \frac{f''(\xi)h^3}{6}

    (Y)
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