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Use areas to evaluate ∫

  1. Feb 25, 2013 #1
    The Question

    Let f(x) = |x|. use areas to evaluate ∫(-1,x)f(t)dt for all x. use this to show that d/dx∫(0,x)f(t)dt = f(x)

    not sure hot to evaluate the integral using area when i dont know what f(t) is...
     
    Last edited: Feb 25, 2013
  2. jcsd
  3. Feb 25, 2013 #2

    Dick

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    f(x)=|x|. So f(t)=|t|.
     
  4. Feb 25, 2013 #3

    epenguin

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    You know what f is.

    You know what mathematical symbolism is. :wink:
     
  5. Feb 25, 2013 #4
    As to your next question, I'm not sure what you mean by "use areas", but I recommend that you draw out what f(x) looks like. Then, see if you can spot an elementary shape the area under -1 to 0 looks like and the same with the area under 0 to x.
     
  6. Feb 25, 2013 #5
    oh.. ya thats kinda obvious now that you point it out :P thanks :)

    ya thats basically what using the area is :P just didnt clue in to what f(t) was :P
     
  7. Feb 25, 2013 #6
    so heres my attempt:

    http://img692.imageshack.us/img692/3284/graphed.png [Broken]
    with f(x) = |x| so f(t) = |t| graphed above, and the area from -1 to x would be

    (1/2)t2 -1/2 = ∫(-1,x)f(t)dt, so

    d/dx(∫(0,x)f(t)dt) = f(x)

    d/dx(1/2x2) = |x|

    x = |x|

    that seem correct?
     
    Last edited by a moderator: May 6, 2017
  8. Feb 25, 2013 #7
    Yes, it's correct, but I have to be picky in how you showed it. You should start with the LHS of what you want to show, ie, d/dx∫(0,x)f(t)dt, and simplify it to f(x). Like this:

    LHS
    = d/dx∫(0,x)f(t)dt
    = d/dx((1/2)x^2)
    = x
    = |x|.....[since x >= 0]
    = f(x), which is what we wanted to show. □
     
    Last edited: Feb 25, 2013
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