Dull Men's Club

Integration: just think about driving somewhere in your car. How long it takes you to get there depends on how fast you’re going as well as the total distance you need to travel. This is easy to figure out if you’re going exactly one constant speed for the entire trip.

But what if you’re driving in the city and you’re speeding up and slowing down all the time? Maybe sometimes you even need to go in reverse which takes you further away from the destination! So the problem you have is that your speed is changing throughout the trip (sometimes even going negative when you’re driving in reverse) and you still want to somehow figure out how long it takes.

Perhaps you might have thought of a straightforward (albeit tedious) way of getting the answer: record your current speed and the current time in a notebook every so often and then multiply each speed by the time interval to get all these small slices of distance travelled, then add them all up at the end. Congratulations, that’s the idea behind integration!

There’s one problem though: your speed might be changing throughout each time interval so the answer you get has all these mistakes that just add up to give you an inaccurate result! Solution: make all the time intervals smaller! This way each time interval includes a more accurate speed, so there will be less errors when we add them all up. Thus the theory of integration is that we can make the intervals arbitrarily small (as small as we want) to get an answer as accurate as we want all the way down to infinite intervals of infinitesimal length which ought to give us the exact answer (and it does)!

Differentiation (calculation of derivatives): the opposite of integration. This takes an entire trip in the car and gives us the ability to calculate our exact speed at any instant in time. Unlike how we might calculate an average speed by looking at the total distance travelled over a time interval and dividing it by that length of time, a derivative gives us the exact speed at a moment in time!

Anyway I hope those two rough explanations help illuminate things a bit for you. Integration and differentiation (calculus) are actually way more useful than just for calculating speeds and distances. For example, the same ideas can be used to calculate the area of an irregular shape (divide it up into squares and add them all up, making the squares smaller helps reduce the errors around the edges) or the volume of an irregular container! Or the slope of a hill at any point on its surface! Or perhaps the lowest point of a valley (using a technique called gradient descent).

That last one is actually very commonly used in artificial intelligence as a training technique. There you’re trying to find the minimum point in some higher dimensional data, according to some rules you have. With gradient descent you use differentiation to find the slope and then you take a small step along the slope towards that minimum point, then repeat!

I understood them 10 years ago but now they are gibberish to me.