The connection between the area under a curve and the Fundamental Theorem of Calculus (FTC) is really interesting! It links two important ideas in math: differentiation and integration. Let’s break it down into simpler parts:

1. Part 1 of the FTC:

This part explains that if we have a smooth function, called $f$, over a certain range from $a$ to $b$, we can create a new function, $F(x)$, that represents the area under the curve from $a$ to $x$. We write it like this:

$F(x) = \int_a^x f(t) dt$

What this means is that if we take the derivative of $F(x)$, which we can think of as how fast the area is changing at a certain point $x$, we get back our original function $f(x)$.

So, if we know how much area is under the curve at different points, we can understand the height of the original function at those points.

2. Part 2 of the FTC:

This part helps us find definite integrals, which is just a fancy way of saying we want to know the area under the curve of $f$ between two points, $a$ and $b$.

To do this, we can simply calculate:

$F(b) - F(a)$

Here, $F$ is any antiderivative of $f$. Instead of going through lots of complicated math, we can just plug in the endpoint values into $F$.

Overall, the FTC ties together the ideas of area and how things change, which makes calculus really cool and useful!