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Prove f is integrable using six subintervals and evaluate.

  1. Apr 11, 2010 #1
    1. The problem statement, all variables and given/known data
    Define f as: f(x)= 2 if 0[tex]\leq[/tex]x<1
    f(1)=0
    f(x)= -1 if 1<x<2
    f(2)= 3
    f(x)=0 if 2<x<3
    f(3)=1
    Prove f is integrable using six subintervals and find the value of [tex]\int[/tex]f(x) dx


    3. The attempt at a solution
    Let P={0, 1-h, 1+h, 2-h, 2+h, 3-h, 3} where 0<h<1/2
    I just need help proving [tex]\int[/tex]f(x) dx=1.
    I can prove I[tex]\geq[/tex]1, but I'm having trouble proving J<1+[tex]\epsilon[/tex]
    I have S[tex]_{P}[/tex]=2(1-h)+2(2h)+(-1)(1-2h)+3(2h)+0(1-2h)+h= 2+9h
    but now I don't know how to get that to less than 1+[tex]\epsilon[/tex]
     
  2. jcsd
  3. Apr 11, 2010 #2

    Mark44

    Staff: Mentor

    I would split the interval up this way:
    P = {0, 1 - h, 1, 1 + h, 2 - h, 2, 3}

    I'm basically ignoring the discontinuities at 2 and 3, since they aren't going to contribute anything to the integral.
     
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