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Homework Help: Improper Integral question.

  1. Mar 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider the function f(x)=1/(x^p).

    When p>1, the integral of 1/(x^p) from 1 to infinity converges.


    i) For what values of p does the integral of 1/(x^p) from 0 to 1 converge? (0<p<infinity, p does not equal 1).

    ii) Confirm the answer by re-writing the integral of 1/(x^p) from 0 to 1 in terms of area and an integral in terms of y. Comment on any symmetry/asymmetry that this relation demonstrates).

    I did a question similar to this but it was much simpler. I have no idea how to even start with this question. PLEASE HELP!!


    3. The attempt at a solution
     
  2. jcsd
  3. Mar 9, 2012 #2

    LCKurtz

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    Just calculate ##\int_\epsilon^1 x^{-p}\, dx##, let ##\epsilon \rightarrow 0^+## and see which ##p## values work.
     
  4. Mar 9, 2012 #3
    Is this your answer to part one? If so, consider that these values will make the integral diverge, not converge. Also, please show us your work.

    [tex]\int_{0}^{1} x^{-p} dx = ? [/tex]
     
  5. Mar 10, 2012 #4
    No that is not the answer to part one. Everything I have posted is part of the question and I have no clue how to get started and finish the question.
     
  6. Mar 10, 2012 #5

    LCKurtz

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    Did you read post #2? That gives a clue.
     
  7. Mar 10, 2012 #6
    On the interval [0,1], which is where we are evaluating the integral, are there any points of discontinuity for values of x?

    If so, then the integral is improper, and you must replace that x-value with a variable, then take the limit as that variable approaches the x-value from either the right or the left. Your book will have a specific theorem or example of an improper integral.
     
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