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Find n for the shaded region

  1. May 16, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the value of n so that the area of the shaded region (refer to attached picture) in the following diagram is a) 50% of the area of the unit square b) 80% of the area of the unit square


    2. Relevant equations
    Definate integral properities, fundamental theorem of calculus



    3. The attempt at a solution
    My teacher says this is an easy question, but I cannot seem to solve it. My first guess was to take the integral of the two functions and then use the percentages as the answers for the integral, solving for n in each case (in this case I made 50% = 1/2 and 80% = 4/5). This did not get me very far for I came up with square roots and fractions in my answers when they should be simply be one number answers. Am I doing this question right through this method, or is this the wrong procedure and do I need to do something different to find the correct answers? Any help would be very welcome, thanks in advance. PS: sorry for the bad picture :tongue:
     

    Attached Files:

  2. jcsd
  3. May 17, 2009 #2

    Mark44

    Staff: Mentor

    The picture has been pending approval for several hours. Can you describe the region in words?
     
  4. May 17, 2009 #3
    There are two functions, y = x^n and y = x^1/n located in region I of the graph. The area to be found is the area in-between these two functions, and runs from their point of intersection at (0,0) to the other point at (1,1). y = x^n is located below y = x^1/n (though i'm sure you probably already figured that out :tongue:).
     
  5. May 17, 2009 #4
    [tex]A=\int_0^1\int_{x^n}^{x^{1/n}} dydx[/tex]
    (depending if n>1 or n<1)

    Evaluate the integral then solve for n. I get:

    [tex]\frac{n}{n+1}-\frac{1}{n+1}=A[/tex]
     
    Last edited: May 17, 2009
  6. May 17, 2009 #5
    I understand the last part, but I don't know how to get (n-1)/(n+1) from that integral you used. Unless you mean finding the integral of x^1/n - x^n over [0,1]?
     
  7. May 18, 2009 #6
    Hi. Didn't notice you replied until now.

    << solution deleted by berkeman >>
     
    Last edited by a moderator: May 18, 2009
  8. May 18, 2009 #7
    Thanks squidsoft, I managed to solve that one.
     
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