How Does The Fundamental Theorem of Calculus: Redefining Integration Work?

The Fundamental Theorem of Calculus shows how integration and differentiation are best friends, one can undo the other.

Imagine you're on a bike ride, and your speed changes all the time. If someone tells you exactly how fast you were going at every moment, you could figure out how far you traveled over the whole trip. That’s like integration, adding up little pieces of distance from each small part of your journey.

Now, think about a stopwatch. If you know how far you've gone after every second, you can find out your speed at any point by seeing how much distance changed in that moment. That’s like differentiation, looking at how things change bit by bit.

Here’s where it gets fun: if someone gives you a speed chart (your changing speed over time), you can use integration to find your total distance, just like knowing where you ended up from all those little speed bits.

But if you have the final distance traveled, and you want to know how fast you were going at any moment, you do the reverse: differentiation. It’s like having a map of your journey and figuring out your speed from the changing points on that map.

So in a way, they’re like best friends, one helps you go from small changes to total amounts, the other goes the opposite way!