I know it’s the “area under the curve”, but what does that mean exactly? Is there a physical or tangible way to explain it?
I understand that a derivative is rate of change at a specific point, and something like acceleration is rate of change of speed. But how can I visualize that speed is the “integral” of acceleration? What does that mean, and how does it relate to the area underneath?
In: 172
An example:
If you can integrate a formula of a flow rate, you get the total volume.
Real world most integration is done already and you won’t need it. because you deal with data points and rarely theoretical functions. Had to learn so much integration as an engineer but never had to do any in my industry.
It’s a way of continuously adding stuff.
Like if I give you a bucket of 1 gal of water, then a bucket of 2 gal, then a bucket of 3 gal, you can just do 1+2+3. But if I turn on a hose and the flow from hose is continuously increasing, then you need to add this up using an integral.
This is a really nice question which has highlighted a lack of clarity in my thinking. Having read through the comments for ultimate simplicity I am think I am I correct to say the integral is “what that achieved“.
so Integral of acceleration tells you the change in speed that particular acceleration achieved. Integral of speed tells you the distance covered (change in position that the speed achieved). Integral of the vol/second of water into the bucket is what that spray achieved (put x volume of water into the bucket). Integral of paycheck is what someone giving you money achieved over the period, ie your gross annual.
I always found using location the easiest.
The derivative of location (change in location over time) is velocity.
The derivative of velocity (change in velocity over time) is acceleration.
Integration is just going back the other way.
The sum of acceleration over time gives you the velocity at a given time.
The sum of velocity over time gives you the location at a given time.
The cumulative amount
Eg if the graph shows the number of litres per minute coming from a hose over time, the integral is the total amount of water that has been sprayed
And as you say the differential is whether, at any given time, the flow rate is increasing or decreasing
Integrals are adding up and combining all the (infinite) pieces into a whole.
It kinda matches up with the plain-english indea of integrating things, meaning to combine them all together.
* If you integrate a car’s speed (over time), you calculate how far it went, becuase you are adding up all the infinitesmal bits of motion it did at every infinitesmal moment in time.
* If you integrate a force (over time) you calculate how much it cause things to move (the ‘impulse’ or ‘change in momentum’), because you add up all the tiny microseconds of force being applied, together to get a total amount of movement you’ve caused
* If you integrate a circle (across a path) then you calculate the volume of a cylinder, because you add up the infinite number of circles that a cylinder is (or can be thought of as being) made up from
The ‘area under the graph’ is a convenient way to represent the number we are calculating in these cases, but those examples are the actual physical things the area represents.
[We can do integrals in more abstract cases too, but those are some actual physical instances.]
derivative is the ratio between output and input
integral is the product of input and output
you need limits to make sense of both though.
Speed is the integral of acceleration and position is the integral of speed. If you think of a rate of change as “subtracting” something concrete (where you actually are on the path between points A and B) to arrive at something more abstract, then the integral is doing the opposite of that – putting the concrete back in.
The most pressing need to compute integrals, historically, has been artillery firing tables. That’s because, given a gun of specified size, a propellant charge of specified power, and an artillery shell of specified weight, it’s pretty easy to determine how fast the shell is going when it leaves the cannon.
But firing artillery at an enemy isn’t about how fast the shell is going, it’s about how far it goes before it hits the ground. Because you’re trying to hit a particular position where the enemy is. So the integral is the act of taking the abstract rate of change in position of the shell over time (the speed) and computing the concrete change in its position altogether (where it actually lands.) That this needed to be approximated (there aren’t analytical solutions for most useful integrals) at increasing levels of accuracy stimulated the invention of the first electronic computers.
It’s like a running total. It’s a summation of the results of plugging every possible input value into the function between two bounding values.
In your example of linear motion, the integral is analagous to the distance covered between the two points in time. If the original function describes speed (distance over time), then differentiating it measure the rate of change of the rate of change of distance over time – the acceleration. You are adding an extra degree of time.
By integrating, you’re removing a degree of time; if you do this with speed, you’re just left with the distance.
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