Let's say you want to download a perfectly legal, uncopyrighted 1.2GB file using bittorrent. There is a graph at the bottom of the screen in your downloading software that shows the number of MB per second downloaded. When the download is going well, the graph line is up high, meaning you're downloading a high number of megabytes every second. When the line is near the bottom of the screen, that means you're not getting very much data each second and it's gonna be a long time before that file is downloaded.

With me so far?

Here's the big question: what is it about this graph that shows how much, total, has been downloaded so far?

It's not the height of the graph at any one point; that's showing how fast data is coming down the pipe.

Suppose you were getting 5MB per second for 100 seconds. Then you would have 500MB downloaded. Or you could get 1MB per second for 500 seconds, and still end up with 500MB total downloaded. A little bit per second for a long time, or a lot per second for a short time. You just multiply the height (MB per second) by the length of the graph (seconds) to get the total downloaded (in MB).

In other words, you calculate the area under the graph to get the total amount downloaded.

The area under the graph of download rate gives the total amount downloaded.

The area under the graph of the rate gives the total.

That sentence is what calculus is all about. "Taking the integral" just means calculating what the total area under the curve is at each moment in time. Newton's big insight was that the graph of the rate and taking integrals were related in just this way-- If you want a graph of how much total has been downloaded so far and all you have is a graph of how much is being downloaded each second, you can "take the integral."

All of the rest of first year calculus is just tricks and methods to calculate this more quickly, or to go in the opposite direction and get the graph of MB per second from the graph of how much total has been downloaded at each point. That's called "taking the derivative."

With me so far?

Here's the big question: what is it about this graph that shows how much, total, has been downloaded so far?

It's not the height of the graph at any one point; that's showing how fast data is coming down the pipe.

Suppose you were getting 5MB per second for 100 seconds. Then you would have 500MB downloaded. Or you could get 1MB per second for 500 seconds, and still end up with 500MB total downloaded. A little bit per second for a long time, or a lot per second for a short time. You just multiply the height (MB per second) by the length of the graph (seconds) to get the total downloaded (in MB).

In other words, you calculate the area under the graph to get the total amount downloaded.

The area under the graph of download rate gives the total amount downloaded.

The area under the graph of the rate gives the total.

That sentence is what calculus is all about. "Taking the integral" just means calculating what the total area under the curve is at each moment in time. Newton's big insight was that the graph of the rate and taking integrals were related in just this way-- If you want a graph of how much total has been downloaded so far and all you have is a graph of how much is being downloaded each second, you can "take the integral."

All of the rest of first year calculus is just tricks and methods to calculate this more quickly, or to go in the opposite direction and get the graph of MB per second from the graph of how much total has been downloaded at each point. That's called "taking the derivative."

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