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How to integrate this?

  1. Jan 20, 2015 #1
    1. The problem statement, all variables and given/known data

    photo of the problem

    g(x)=∫ƒ(t) dt from -5 to x

    ƒ(t) = (0 if x < -5
    5 if -5≤x<-1
    -3 if -1≤≤x<3
    0 if x≥3)

    (a) g(-8) = 0
    (b) g(-4) = 5
    (c) g(0) = ?
    (d) g(4) = ?

    2. Relevant equations

    ∫ƒ(x) from a to b = (f'(b)-f'(a))
    fundamental theorem of calculus

    3. The attempt at a solution

    I was able to get that g(-8) = 0 because plugging -8 into the upper limit and meant x was less than -1 and gave me with f(t) equaling 0 and the anti derivative of 0 is 0. Now i got g(-4) = 5 because the antiderivative of 5 is 5x and plugging in -4 → x and -5→x got me 5. However when i do the same for c and i get -15 by plugging in 0 to x making the f(t) = -3 . The last 0 i thought the anti derivative of 0 is 0 but it doesn't take either of those answers. I need guidence.
    Last edited by a moderator: Jan 21, 2015
  2. jcsd
  3. Jan 21, 2015 #2


    Staff: Mentor

    This is NOT what the FTC says!
    Much better! Thank you.

    Sketch the graph, if you haven't already done so. If you have the graph, this is a very simple problem.
    For c) g(0) = ##\int_{-5}^{-1} f(t)dt + \int_{-1}^0 f(t)dt##
    The first integral is positive, since the graph of f is above the horizontal axis. The second integral is negative, because the graph is below the hor. axis.

    For d), the approach is similar.
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