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Area Between Curves

  1. Oct 12, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the area bounded by the curves y=x^2 and y= 2 - x^2 for 0 ≤ x ≤ 2.


    2. Relevant equations

    ∫top - ∫bottom


    3. The attempt at a solution

    ∫(2-x^2)dx - ∫x^2dx

    What I'm confused about is that the two equations only cross on [-1,1] so within the interval of the problem I only have an enclosed area on [0,1]. But the problem asks for the area on [0,2]. How do I reconcile the differing intervals?
     
  2. jcsd
  3. Oct 12, 2011 #2

    LCKurtz

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    If you draw the vertical line x = 2 it gives a right boundary just like x = 0 gives the left boundary. Your curves cross so you have to do it in two parts.
     
  4. Oct 12, 2011 #3
    OH! I think I see that now.

    So I'll have:

    [∫(0→1)(2-x^2)dx - ∫(0→1)x^2dx] + [∫(1→2)(2-x^2)dx - ∫(1→2)x^2dx]

    Basically the area between the curves on [0,1] plus the bits hanging off on [1,2].

    A = 4/3 un^2

    I knew there was something I was missing and it's been a couple of weeks since we did that.

    Thanks for the helps!
     
  5. Oct 12, 2011 #4

    LCKurtz

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    Your integrand is always y-upper - y-lower. Check that on the interval [1,2].
     
  6. Oct 12, 2011 #5
    Oh! Yep. I forgot that my lines crossed.

    One step at a time.... :)

    A = 4 un^2

    Thanks again.
     
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