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?

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?

Reagan Madden

Beginner2022-06-23Added 15 answers

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 $a<b,$, 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 $y=f(x),$, 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 ${\int}_{a}^{b}f$

$=F(b)-F(a).$

4. The actual area enclosed by the curve $y=f(x),$, the x-axis, and the lines $x=a$ and $x=b$ is given by ${\int}_{a}^{b}|f|,$, which does not equal $F(b)-F(a).$

5. In your Question, the value evaluation $F(0)=-1$ does not correspond to an area.

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 $a<b,$, 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 $y=f(x),$, 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 ${\int}_{a}^{b}f$

$=F(b)-F(a).$

4. The actual area enclosed by the curve $y=f(x),$, the x-axis, and the lines $x=a$ and $x=b$ is given by ${\int}_{a}^{b}|f|,$, which does not equal $F(b)-F(a).$

5. In your Question, the value evaluation $F(0)=-1$ does not correspond to an area.