FundamentalTheoremofCalculus(Part2)
                     Created by
               Barbara Forrest and Brian Forrest
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      FundamentalTheoremofCalculus(Part2)
      Example: Evaluate         Z
                                 2
                                    3
                                   t dt.
      Observation: If            0
                                   Z x 3
                            G(x) =     t dt,
                                    0
      then                         Z
                                     2
                                       3
                            G(2) =     t dt.
      WeknowfromtheFTC1that         0
                              G′(x) = x3
                             3
      so Gis an antiderivative of x . Hence there exists a constant C such
      that                    Z
                                x         4
                                  3      x
                       G(x) =     t dt =    +C.
                               0          4
      Question: How does this help us?
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      FundamentalTheoremofCalculus(Part2)
      Note: Wehavejustseenthat
                       G(x) = Z xt3dt = x4 +C
                                0         4
      for some constant C ∈ R.
      However, we also know that
                       Z 0 3             04
                   0 =    t dt = G(0) =     +C=C
                        0                4
      so                        Z x 3      x4
                         G(x) =     t dt =    .
                                  0         4
                       Z 2                4
      Finally,             3             2
                          t dt = G(2) =     =4.
                        0                 4
      Question: Did we really need to find C?
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    FundamentalTheoremofCalculus(Part2)
    KeyObservation:
    Let F and G be any two antiderivatives of the same function f. Then
                   G(x) = F(x)+C.
    Let a,b ∈ R. Then
           G(b)−G(a) = (F(b)+C)−(F(a)+C)
                     = F(b)−F(a).
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