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Thread: Clarification of antiderivative evaluated at a (F(a))

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

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

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

    Thanks
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  2. #2
    MHF Contributor

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