How exactly was the integral derived? Like similarly to how the difference quotient explains where the derivative came from, what can we use to explain the origins of the integral? Like how does evaluating an antiderivative of some function between two bounds equal the area under that curve?
I guess what I am trying to ask is what went through the minds of Leibniz/Newton that made the integral just make sense.
An integral can be thought of as an infinite sum of infinitesimal summands. This is the origin of the $\int$ symbol (introduced by Leibniz).
As you said, it can be interpreted as the area under a curve between two limits $a$, $b$. This interpretation stems from the fact that the area under a curve can be approximated by very thin rectangles like so:
Split the interval $(a,b)$ into $n$ sub-intervals, each of size
$\Delta x = \frac{b-a}{n}$
Let $y=f(x_i)$ be the height of a rectangle with width $\Delta x$, for $i=0,1,2,...,n$ (note $x_0 = a$ and $x_n = b$). Then this rectangle will be a small part, $A_i$, of the total area under $y = f(x)$ between $a$ and $b$.
$ A_i = f(x_i) \cdot \Delta x$
If we add these parts of the area together, we get the total area between $a$ and $b$:
$\displaystyle\sum_{i=0}^{n} A_i = \displaystyle\sum_{i=0}^{n} f(x_i) \cdot \Delta x \approx Total \space Area$
Since the rectangles only approximate the area, there will be an error which decreases as we increase $n$. If we take the limit of the sum as $n \to \infty$, the error disappears. In fact, this is how the definite integral between $a$ and $b$ can be defined:
$\int_a^b f(x) dx = \lim_{n \to \infty} \displaystyle\sum_{i=0}^{n} f(x_i) \cdot \Delta x$
The image shows how the sum converges to the definite integral as $n \to \infty$.