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Area Under a Curve With a Constant, k

  1. Mar 3, 2009 #1
    1. The problem statement, all variables and given/known data

    Let f be the function given by f(x) = kx^2 - x^3, where k is a positive constant. Let R be the region in the first quadrant bounded by the graph of f and the x-axis.

    Find all values of the constant k for which the area of R equals 2.

    3. The attempt at a solution

    So I've pretty much decided that we're comparing two terms here: the kx^2 and the x^3 term. If the x^3 term is bigger than the former, than voila we have our limits of integration!

    I understand that squaring a number between 0 and 1 will make a smaller number... and cubing that same number makes an even smaller number. so, from 0 < x < 1, the x^2 term is bigger than the x^3.

    I tried integrating the function f(x) from 0 to 1 and got a k value of 27/4. This leads me to think that there are infinite limits from x = 0 to x = b where I can get a region of 2.

    Hints please

    Thanks
     
  2. jcsd
  3. Mar 3, 2009 #2

    Avodyne

    User Avatar
    Science Advisor

    Leave k as an arbitrary constant for now. What is the area of the region R? Your answer will be a function of k. Set that function equal to 2, and solve for k.
     
  4. Mar 3, 2009 #3
    Your limits of integration are not correct. It's bounded in the first quadrant so find where the function is in the first quadrant. Hint: Look at the roots, and why is know about cubics.
     
  5. Mar 3, 2009 #4
    If I leave k as an arbitrary constant, I get the following:

    integration of f(x) = 2

    (k/3) x^3 - (1/4) x^4 + C = 2

    The "C" value is throwing me off. I can't integrate correctly without the limits of integration. The limits of integration depend on the value k.

    I understand that my limits are wrong. I'm saying that depending on the value of k, the limits of integration change from 0 to b, where b is some positive number. I'm looking at the roots kx^2 and x^3 but i can't ascertain any other information...

    Thanks for the help so far but I'm still a little stumped
     
  6. Mar 3, 2009 #5

    Mark44

    Staff: Mentor

    Your integrand is kx^2 - x^3. What are the x-intercepts of this function? Those will be your limits of integration. Your integral is a definite integral, so the constant C is unnecessary.
     
  7. Mar 3, 2009 #6
    Dang. That was what I was looking for...

    The x intercepts of f(x) I found was x = 0, x = k.

    I integrated f(x) with lower limit x = 0 and upper limit x = k .. and equaling this integration to 2.

    I get a k value of 24^(1/4). Is this correct?
     
  8. Mar 3, 2009 #7

    Mark44

    Staff: Mentor

    Yep.
     
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