eli5 why the area under a curve(or any line) is important
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If you have a graph which shows speed over time the area under it is the distance travelled and the gradient is the acceleration.
Because that often gives you important information about the system you’re graphing. As a common example, if time is on the x axis then area under the curve is like “the cumulative (y) so far”.
So like
* if you have time on the x axis and income rate on the y axis, the area under the line is the total money you have made.
* if you have time on the x axis and speed on the y axis, the area under the curve is your total distance travelled up until that time
Here’s an actual one from my chemistry Grad School experiments. I was doing a chemical reaction that produced water. The machine told me the rate that water was being produced. I could make a graph of this rate on the y axis and time on the x axis, and the area under that line was the total water produced so far (and water was a product, so from that I could calculate how much of the chemical reaction I was studying had taken place under whatever conditions I was using).
The integral can tell you a lot of information about whatever you are dealing with. For example if you have a graph that shows an objects speed as a function of time the area under the curve will show how far from an arbitrary point you have traveled for any X value. Basically if a graph measures a change in something than the area under the curve will show the total for any value.
“The area under the curve” rarely matters; specific cases in geometry. But that area represents result of an integral. Integral is a special function that takes a function (instead of just a variable) on input and produces another function on output. And integrals, alongside with derivatives (which do the same only in opposite direction) are very important and useful, changing “rate” into “value”
Take accelerometers, small chips, often found in phones, drones etc. They measure acceleration (including gravitational). In phones they’ll usually just change the screen orientation. But in drones – integral of acceleration is speed. Integral of speed is position. So passing the output of accelerometers through integration twice (and knowing the drone was lying immobile on the ground in the beginning) they can tell the drone where it is, and so tell it how to return to the starting point, or hold position, or follow a specific path.
In the opposite direction, your car’s speedometer runs a differential on the input of odometer; position/distance->speed.
Let’s add there’s a whole bunch of other integrals, not just “surface under the curve”. There’s length of the curve, there’s flow through a loop in a field (evocative: how much water in a river will flow into a pipe depending on its shape and orientation; practical: engineering stuff about magnetic fields – electromagnets, electric motors), volume integral (mass of a solid carved out of uneven density material), and a bunch of others so obscure I forgot them.
That’s not a specific definition. It’s an expression that means ‘for us right here right now this is the important data.’
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