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Math Help - Clarification of antiderivative evaluated at a (F(a))

  1. #1
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    Clarification of antiderivative evaluated at a (F(a))

    Hi, given \int_{a}^{b}{f(x)dx}=F(b)-F(a), I simply wanted to clarify what does F(a) on its own actually tell us? Is it the cumulative signed area

    • from 0 to a or
    • from -\infty to a or
    • from arbitrary constant c to a where c varies by choice of f(x) or
    • something else?

    Thanks
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  2. #2
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    Since F(x) is an anti-derivative of f(x) which can vary by an added constant, F(a) can be interpreted as the area from some arbitrary constant c to a where c varies by choice of additive constant, NOT f(x).

    For example, if f(x)= x^2 then an anti-derivative, F(x), is of the form \frac{1}{3}x^3+ C where C can be any constant. If, say, C= 0 then F(a)= \frac{1}{2}a^2 is the area under the curve y= x^2 from 0 to a. If we were to take C= 9, then F(a)= \frac{1}{3}a^2+ 9 is the area under the curve y= x^2 from -3 to a.

    (Remember that "area under the curve" is one possible interpretation of the anti-derivative. It isn't really correct to say that an anti-derivative is an area.)
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