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Find the limit a such that the definite integral equals 8

  1. Mar 16, 2012 #1
    Find the limit "a" such that the definite integral equals 8

    1. The problem statement, all variables and given/known data

    Hello!

    the problem ask me to find for which a in R, does this holds:

    2
    ∫ abs(2x - x^2) = 8
    a

    3. The attempt at a solution

    Well, I see that:

    f(x) { if x > 2, then f(x) is negative; if x < 2, then f(x) is positive.

    I rewrited the integral and got: (the limits are those inside the brackets)

    2
    ∫ (-2x + x^2) + [2,a]∫ (2x - x^2) = [-a,2] - ∫ (2x - x^2) + [2,a]∫ (2x - x^2) or
    -a


    a
    - ∫ (2x - x^2)
    -a

    I know that in such cases, odd functions cancel. I'm left with the even function. I continued and got:

    (2/3) * a^3

    and since the definite integral should equal to 8, I just solved for "a" and got:


    (2/3) * a^3 = 8 or

    a = cuberoot(12).

    But when I wanted to check if (2/3) * (12)^1/3

    I didn't got the expected 8.

    ...

    So it must be wrong.

    I don't see the mistake in my answer.

    Any suggestions?

    Thanks.
     
  2. jcsd
  3. Mar 16, 2012 #2
    Re: Find the limit "a" such that the definite integral equals 8

    You made a mistake where you stated f(x)(i.e 2x-x^2) is positive for x less than 2.

    Plug in x = (-1)
     
  4. Mar 16, 2012 #3
    Re: Find the limit "a" such that the definite integral equals 8

    hello!

    You are right. Yeah, I should have said from 0 <= x <= 2.

    0
    ∫ (-2x + x^2) + [0,2]∫ (2x - x^2) = [-a,0] - ∫ (2x - x^2) + [0,2]∫ (2x - x^2)
    -a

    And then I ended with a cubic polynomial:

    a^3 + 3a^2 -20 = 0

    Which has negative discriminant...

    and its real solution , which is

    a = 2

    Doesn't work...

    :(
     
    Last edited: Mar 16, 2012
  5. Mar 16, 2012 #4
    Re: Find the limit "a" such that the definite integral equals 8

    Your expression is nearly right.You have made a small error.

    (Hint:in the expression coeffient of one term(either a^3 or a^2 term should come up negative)
     
  6. Mar 16, 2012 #5
    Re: Find the limit "a" such that the definite integral equals 8

    Solving the sum is pretty easy.

    Since the graph of abs|f(x)| will lie always above x axis, amd the integral fetches a positive value a<2 (cause if a>2 integral of any positive function for a to 2 will be negative)

    If 2>a>0 you can remove || as it has no meaning.
    If a<0, the you can break the integral from a to 0 integral of [-f(x)] {why did i multioly with minus?} and from 0 to 2[integral of f(x)] {why no minus here?} .

    Solve the sum to be equal to 8.
     
  7. Mar 16, 2012 #6

    SammyS

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    Re: Find the limit "a" such that the definite integral equals 8

    attachment.php?attachmentid=45178&stc=1&d=1331939495.gif

    The above is the graph of f(x) = |2x - x2| .

    For 0 < x < 2 , f(x) = 2x - x2 .

    For all other values of x, f(x) = x2 - 2x .

    By the way, f(x) is never negative.

     

    Attached Files:

  8. Mar 17, 2012 #7
    Re: Find the limit "a" such that the definite integral equals 8

    Apparently I commited a mistake at the beginning (and all of the time) when I wanted to check the value of a. You see in the quote when doing the substituion

    (2/3) * (12)^1/3

    It is supposed to be (2/3) * (12), which is equal to 8. For some reason I was not canceling the cube root of 12.

    But still I my first answer was sloppy. Actually the suggestions (specially the graph) helped me a lot! I tried solving it again and got the answer correct!

    Thank you!
     
  9. Mar 17, 2012 #8

    SammyS

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    Gold Member

    Re: Find the limit "a" such that the definite integral equals 8

    You're welcome.

    As they say, "A picture is worth a thousand words."
     
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