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Area of the region bounded between two curves with integration by parts

  1. Jan 22, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the area bounded between the two curves
    y=34ln(x) and y=xln(x)

    2. Relevant equations

    Integration by parts: [tex]\int[/tex]udv= uv-[tex]\int[/tex]vdu

    3. The attempt at a solution
    First I found the intersection points of the two equation to set the upper and lower bounds. The lower bound is 0 and the upper bound is 34. My integrand is as follows--

    [tex]\int[/tex]34ln(x)-xln(x) with the limits of integration being from 0 to 34.

    I evaluated the integral using integration by parts, and eventually came up with the following solution...

    [tex]\int[/tex]34ln(x)-xln(x)= 34xln(x)-34x-(x[tex]^{2}[/tex]ln(x)/2) -(1/4)x[tex]^{2}[/tex]

    and evaluated from 0 to 34, the answer is 1156ln(34)-1156-(1156ln(34)/2)+289

    I am not quite sure where my mistake is being made. I verified my answer using a graphing calculator (although that does not absolutely make my answer correct), so if anyone sees where I am making my mistake I would greatly appreciate the help!
  2. jcsd
  3. Jan 22, 2010 #2


    Staff: Mentor

    When you say the lower bound is 0, I don't see how that is possible. From your integral, it appears that you are integrating with respect to x. Neither of your functions is defined at x = 0.
  4. Jan 22, 2010 #3
    check this again

    34xln(x)-34x-(xLaTeX Code: ^{2} ln(x)/2) -(1/4)xLaTeX Code: ^{2}

    one of the sign is wrong
  5. Jan 22, 2010 #4
    My mistake was in the bounds, which should have been from 1 to 34. Thank you for your help!
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