The Fundamental Theorem of Calculus (FTC) connects two important ideas in math: differentiation and integration.

Differentiation is about how things change instantly, like speed. Integration, on the other hand, is about adding up quantities over a range, like how far you’ve traveled during a certain time.

Let’s break this down.

First Part of the FTC

The First Part of the FTC says that if we have a function $f$ that is smooth (continuous) on a section from $a$ to $b$, and $F$ is a function that shows the total amount collected from $f$, then:

$$\int_a^b f(x) \, dx = F(b) - F(a)$$

This formula means that to find out how much we accumulated from $a$ to $b$, we can just look at the values of $F$ at the ends, $b$ and $a$, and subtract them.

Think of it this way: if $f$ is like a speedometer showing how fast you’re going, then $F$ is like the odometer showing how far you've gone. The FTC shows us how distance changes when we consider speed over a period of time.

Second Part of the FTC

The Second Part of the FTC tells us that differentiation and integration are opposite processes. It states that if we find $F$ by adding up the function $f$ from some point $a$ to another point $x$, like this:

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

then when we find the derivative (which tells us the rate of change) of $F$, it gives us back the original function $f$:

$$F'(x) = f(x).$$

This means that at any point $x$, the rate of change (slope) we get from $F$ is equal to the value of $f$ at that point.

In summary, these two parts of the FTC show us that understanding how things add up through integration helps us understand their instant rates of change through differentiation. Even though we might study these ideas separately, they work together in many ways in calculus.