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Areas between Curves problem

  1. Feb 27, 2014 #1
    1. The problem statement, all variables and given/known data
    Let a > 0 be a fixed real number. Define A to be the area bounded between y=x2,y=2x2, and y=a2. Define B to be the area between y = x2, y = f(x), and x = a where f(x) is an unknown function.
    a) Show that if f(0) = 0, f(x) ≤ x2, and A = B then

    int 0-->a2 [y1/2-(y/2)1/2] dy = int 0--> a [x2 - f(x)] dx

    b) Find f(x) under the conditions above are true for all a > 0

    2. Relevant equations

    I guess the equations for the lines, y=x2, y=2x2, y=a2
    and y=x2, y=f(x), and x=a

    3. The attempt at a solution

    The first thing I did was try to find the area A.
    A= int 0-->a2 [y1/2-(y/2)1/2] dy
    A= (2/3)y3/2 - (1/sqrt2)(2/3)y3/2| eval. 0 and a2
    A= (2/3)a3 - (1/sqrt2)(2/3)a3
    A= a3((2/3) - (1/sqrt2)(2/3))
    a= (2/3)a3(1- (1/sqrt2))

    Then I tried to find the area B, but this is where I got confused.
    B= int 0--> a [x2 - f(x)] dx
    B= [(1/3)x3|eval. 0-->a] - [int 0-->a [f(x)]]

    I don't see how I can go any farther, because I don't think I can integrate an unknown f(x). Am I going about this the wrong way?
     
  2. jcsd
  3. Feb 27, 2014 #2

    Simon Bridge

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    The relation does not require you to interpret the unknown f(x).
    Take another look at what you have to show.

    If A=B then the relation drops out. The trick is to verify the other two conditions.
     
  4. Feb 27, 2014 #3
    Are the other conditions f(0)=0 and f(x)≤x2? I don't understand how to verify those, would it have to do with the graph of the region? Since A=B and for A, both of the curves have a point at (0,0) so B must have two points there also?

    And how should I go about part b, do I just solve for f(x)?
     
  5. Feb 27, 2014 #4

    Simon Bridge

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    Yes.

    Yes.

    No.
    A=B just means the areas are the same - the graphs do not have to have points in common for that to happen.

    i.e. what would happen to the RHS if f(x) > x^2

    You should make sure you understand part (a) first, but basically you can separate out the integral in f(x) and use your understanding of the properties of integrals.
     
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