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Integration from first principles

  1. Dec 23, 2007 #1
    Please help me check my solution. Is there a simpler way to do this?

    1. The problem statement, all variables and given/known data

    Integrate x^3 cos x from first principles.

    2. Relevant equations

    Taylor series of cos x
    Sum of powers from 1 to n

    3. The attempt at a solution

    I am dividing the area under the curve from a to b into n strips and then summing up the areas (where the area is x.f(x)). Then express this in terms of n and let n tend to infinity.

    Let Sn = (Sigma) f(x') (delta)x
    f(x) = x^3 cos x
    f(x) = x^3 - x^5/2! + x^7/4!

    x' = j . (delta)x

    Insert values from 1 to n, and then group

    Sn = [(delta)x]^4 . (1^3 + 2^3... + n^3)
    + [(delta)x]^6 . (1/2!) . (1^5 + 2^5 ... + n^5)
    + [(delta)x]^8 . (1/4!) . (1^7 + 2^7 ... + n^7)

    Delta x = b / n, where b is the upper limit for integration

    which yields 1/4 b^4 + 1/12 b^6...
  2. jcsd
  3. Dec 23, 2007 #2
    I'm not really sure what "first principles" means, but shouldn't Delta x = (b-a)/n instead of just b/n?

    Also you could do it with integration by parts, but I don't know if they want that or not.
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