The area under a curve, however the curve may go as long as it doesn't double-back then it's good for figuring the area between it and the $x$-axis, and is derived similarly to the tangent, as follows. The area between a function and the $y=0$ line ($x$-axis) is named the integral of that function.

First we define the area between a curve and the line which is the zero-ordinate for all $x$,
on an interval of abscissa,
to be some function, $F$, of the interval, $[x_1, x_2]$, and the curve, as $F(x_1, x_2, f)$.
The change in $F$ for a given change $b$ in the interval defined, chosen at the upper end of the interval ($x_2$).
So, to use the Greek letter Delta
(for the root of *difference*
$\delta \iota \alpha \phi \omicron \rho \alpha$
)
to symbolize the difference of the quantity:
$$
\Delta F = (b)\times(f(x_2+b))
$$
This is the formula for an area which is a rectangle of width $b$ and height $f$, and as long as this increment is not at a point where the difference of $f(x_2+b)$, the height of the addendum area, and $f(x_2)$, the height for the adjacent $b$-width portion,
is infinite then this construction is meaningful.

So, the change in the area per change in interval is proportional to that change, ($\Delta x = b$), and the difference between $f(x)$ and $y=0$ at $x=x_2$ (which is simply $f(x_2+b)$). Below, the area between the low-point and zero is constant and left out.

The difference in area acquired by extending the interval boundary by $b$ is simply the height of the function at $(x_2+b)$ times the width, the differential parameter $b$. The declaration that the area is some analytical quantity, calculable from some infinite number of contiguous rectangles, without adding more constraints than exist, we can say the quantity of such area is, in general, a function of the following parameters.

- $f(x)$, the curve making the top boundary,
- $x_1$, the line going from the point $(x_1, 0)$ to point $(x_1, f(x_1))$, as a section of the entire vertical line $(x_1, y)$, making the left boundary,
- $x_2$, defining the right boundary.

The area which could be thought of as a (finite) series of the function in subject as kernel to a summation operating from the interval lower bound in subject to the intervals upper bound all terms multiplied by some differential parameter $k$, as the uniform width of the rectangle heights.

The $k$-width rectangles add up to some area equivalent to the area between the function $f(x)$ and the $x$-axis (the abscissa), for the given interval $[a, b]$. So, the change in that total area as a function of the upper bound of the interval $b$ is the height of the the curve at $b$ so $F(x+b) - F(x) = b f(x) $

For historical placement of this notion, the anti-derivative of $x^{-1}$ was solved by Gottfried Leibniz in 1676, first publisher of the formalism of tangents [1].

The example polynomial used above, $f$, is a truncated sine series, of four times the argument, to sixth degree in the Maclaurin series. And the common jargon today refers to the relationship between the integral and the derivative of a function, the Fundamental theorem of calculus.

[1] Scriba, Christoph J. (1963). The inverse method of tangents: A dialogue between Leibniz and Newton. __Archive for History of Exact Sciences__ 2 (2):113-137.

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