Mar 10 2024
Did you know the fact that the integral calculates the area under a graph is just a mathematical coincidence? When Newton was creating calculus for use in his kinematic model of physics (\(\vec{F}=m\vec{a}\)) he likely had no idea that it was adding up the area under the graph. Here is an alternate way to think about the integral, much more applicable to physics that tries to explain what an integral actually means, and how it applies to physics, in a more intuitive way.
imagine you already know the velocity over time for some object. you have a function describing it at every instant in time that you're interested in, \(v(t)\). how can we construct a function to describe the position?
the first thing you might try is a simpler case. Instead of allowing any velocity function we can imagine, its simpler to just consider a constant velocity, so lets think about the specific example shown below:
in this case its easy to work out where the object is at any point in time just by multiplying, since we always move in the same speed and in the same direction.
to get closer to our goal of understanding continuously changing speeds, lets now consider travelling with this velocity for a certain ammount of time, say one second, and then suddenly changing direction and moving for the same ammount of time.
moving in certain directions for certain ammounts of time lets us calculate the position of the object by adding up the movement in every direction its moved in. this is something we can exploit to find the position when we have constantly changing speed and direction.
we can check the velocity at the current point in time, and move that way for a short duration before checking again to make sure we're going the right way.
but where do we start? it actually doesn't matter! you can start at any position you want and move the object from that starting position (this is what the \(+c\) is) since we are only calculating movement, it doesn't matter where you start, it just shifts everything over by a fixed ammount.
as you can see from the demo above, the time we wait before checking our direction again (\(dt\)) has a big effect on the accuracy of our result, its closer to reality when we check the direction more often, but checking more often means adding up more movements to get our final result.
This process shown above is the main idea behind integration - not the area under a curve. Its just a coincidence that \(\text{speed } \times \text{ time}\) to calculate distance traveled is the same formula as \( \text{length } \times \text{ width} \) for an area when looking at the graph of one dimensional velocity.
The proper definition of the integral allows you to shrink the time between changing direction to "infinitely close" to zero so that you can get a result exactly equal to what the constantly changing velocity would do in the real world.
This is also why the integral symbol looks the way it does, its meant to look like a big "S" meaning sum, since you're adding up lots of tiny directions to work out the large movements they create. $$ \int dt $$