What area does the antiderivative represent?

Evaluating the antiderivative at the value $0$ produces $-1$. Now, I was taught to conceptualize an antiderivative as an area under a curve

1. For $t>a,$ the signed area (i.e., the area in the positive vertical region minus the area in the negative vertical region) $A(t)$ enclosed by the curve $y=f(x),$ the $x$-axis, and the vertical lines at $a$ and $t$ is denoted by the (definite) integral $$\int_a^tf(x)\,\mathrm dx,$$ whose elongated-S symbol stands for Sum (of areas).

2. If $f$ is continuous on $[a,t],$ then the Fundamental Theorem of Calculus says that this integral is computable using the formula $$\int_a^tf(x)\,\mathrm dx=F(t)-F(a),\tag1$$ where $F$ is an antiderivative of $f,$ that is, any of the infinitely many functions whose derivative (slope function) is $f.$ Therefore, if $f$ is continuous on $[a,t],$ then $$\frac{\mathrm d}{\mathrm dt}\int_a^tf(x)\,\mathrm dx=f(t).\tag2$$ Rewriting $(1):$ if $F$ is continuously differentiable on $[a,t],$ then $$\int_a^t\frac{\mathrm d}{\mathrm dx}F(x)\,\mathrm dx=F(t)-F(a).\tag1$$

3. The indefinite integral $$\int f(x)\,\mathrm dx$$ represents every antiderivative of $f,$ if any. For example, \begin{align}\int \frac1x\,\mathrm dx &= \begin{cases} \ln|x|+C_1, &x<0;\\ \ln|x|+C_2, &x>0.\end{cases}\end{align}

   Conceptually, an indefinite integral is not technically the precursor of a definite integral: for example, for \begin{align}g(x)&= \begin{cases} 2x\sin\frac1{x^3}-\frac3{x^2}\cos\frac1{x^3}, &x\ne0;\\ 0, &x=0,\end{cases}\end{align} $\displaystyle\int_{-1}^1 g(x)\,\mathrm dx$ does not exist even as \begin{align}\int g(x)\,\mathrm dx&= \begin{cases} x^2\sin\frac1{x^3}&+C, &x\ne0;\\ 0&+C, &x=0,\end{cases}\end{align} that is, even as $g$ does have antiderivatives on $[-1,1].$

4. Since antidifferentiation and indefinite integration literally reverse differentiation, they don't need to be specified on an interval.

   On the other hand, definite integration is a computation with reference to some interval.

   The Fundamental Theorem of Calculus ($(1)$ & $(2)$ above) shows the surprising intimate relationship between definite integration and differentiation.

5. An antiderivative per se does not describe area, and $F_λ(0)=-1$ means that a particular function whose slope is given by $f$ passes through $(0,-1).$

   Clearly, a single value of an antiderivative is not useful information; the FTC's power comes from making its two evaluations using the same antiderivative of $f:$ $$\int_a^tf(x)\,\mathrm dx=F_λ(t)-F_λ(a)\\\neq F_λ(t)-F_ω(a).$$

6. A pertinent comment by B. Sullivan:

   I just finished reading Steven Strogatz's new book about calculus, Infinite Powers. In it, he lays out the "Three Central Problems of Calculus":

   1. Given a curve, find its slope everywhere. [differentiation]
   2. Given a curve, find another curve whose slope everywhere is that given curve. [antidifferentiation]
   3. Given a curve, find the area under it. [(definite) integration]

7. The actual area enclosed by $y=f(x),$ the $x$-axis, and the vertical lines at $a$ and $t$ is $$\int_a^t|f(x)|\,\mathrm dx,$$ which does not generally equal $F(t)-F(a).$