Craig Mendoza

2022-06-22

What area does the antiderivative represent?
Consider the antiderivative of the function ${e}^{-x},$, which is $-{e}^{-x}.$. Evaluating the antiderivative at the value 0 produces -1.
I was taught to conceptualize an antiderivative as an area under a curve, or a sum of progressively smaller approximate sections.
But clearly, -1 cannot represent the area under under the ${e}^{-x}$ curve from 0 to 0, 0 to $\mathrm{\infty }$, or $-\mathrm{\infty }$ to 0 when you consider that the function is positive for all values of x.
Then what sum or area does the value of the antiderivative of ${e}^{-x}$ actually represent?

Explanation:
Evaluating the antiderivative at the value 0 produces −1. Now, I was taught to conceptualize an antiderivative as an area under a curve
1. An antiderivative F results from performing indefinite integration on a function f.
2. For $a, the signed area (i.e., the area residing in the positive vertical region minus the area residing in the negative vertical region) enclosed by the curve $y=f\left(x\right),$, the x-axis, and the lines $x=a$ and $x=b$ is given by the definite integral ${\int }_{a}^{b}f.$
3. By the Fundamental Theorem of Calculus, the above signed area ${\int }_{a}^{b}f$
$=F\left(b\right)-F\left(a\right).$
4. The actual area enclosed by the curve $y=f\left(x\right),$, the x-axis, and the lines $x=a$ and $x=b$ is given by ${\int }_{a}^{b}|f|,$, which does not equal $F\left(b\right)-F\left(a\right).$
5. In your Question, the value evaluation $F\left(0\right)=-1$ does not correspond to an area.

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