The Fundamental Theorem of Calculus (FTC) shows a deep link between two important math ideas: differentiation and integration. These two ideas help us understand how things change and how to find total amounts.

The Two Parts of the FTC

1. First Part: If we have a smooth function called $f$ between two points, $a$ and $b$, and a related function called $F$, the FTC tells us:

   $$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

   This means we can find the area under the curve of $f(x)$ from point $a$ to point $b$. It connects integration, which finds total amounts, to the idea of change over time.

2. Second Part: This part says that if $f$ is smooth from $a$ to $b$, then the function $F(x) = \int_{a}^{x} f(t) \, dt$ can be changed back into $f$ by taking the derivative. In simpler terms:

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

   So, differentiation and integration are like opposites of each other. The change of the area function gives us back the original function.

Applications of the FTC

• Area Under Curves: The FTC helps us easily find areas, which is really useful in fields like physics and engineering.

• Accumulated Change: It helps us figure out total amounts like distance traveled, total growth, or total earnings, such as calculating total money made from sales that change over time.

In short, the FTC is a key part of calculus. It shows how integration can help us find total amounts, while differentiation helps us understand how things change.