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
Accumulation.
For example if you have a faucet filling a bucket and your function is how open the faucet is, the integral is how much water accumulated in the bucket.
Similarly with speed, if your function is acceleration, the integral is how much speed has built up.
The area under the curve is an accumulation of a bunch of thin vertical rectangles, each of which has a height equal to the function at the corresponding point.
Made me think of the “chemist integration” method: Plot the function, cut it out with scissors, measure on a scale 🙂
Seriously though, if you are OK with acceleration being the “rate of change” of speed, just reverse it: speed is the accumulation of acceleration over time.
It’s a simple multiplication for a constant (area of a rectangle), multiplication with the average for something linear (area of a rectangle plus a triangle), and for anything more complex, well, however the area under the curve grows over time.
If speed is your function. Acceleration would be the derivative, and the distance covered the integral.
If you take the integral of the velocity function, you’ll get the position function as a result.
v = v_0 + at
where v = velocity, v_0 = initial velocity, a = acceleration, t = time
x = x_0 + v_0t + ½at²
Where x = position and x_0 = initial position.
Take the derivative if the position function with t as the variable. You’ll get the velocity function.
It helps to keep track of the units. Velocity is distance per time, acceleration is distance per time squared, position is distance. The units in the position function are: assuming metric
m = m + m/s * s + m/s² * s²
Same with the velocity function
m/s = m/s + m/s² * s
I struggled with calculus until I took physics.
Iirc we used integrals to calculate volumes of different weird shapes. It’s been a long time, but that part sticks out.
Imagine we have a rocket, and the rocket burns its fuel to go faster and faster and faster.
At any one time, it’s moving at a particular speed, has travelled a certain distance, and is accelerating a certain amount.
These things have integral and derivative relationships with each other.
Distance is the integral of speed, and speed is the derivative of distance.
Speed is the integral of acceleration, and acceleration is the derivative of speed.
Antiderivative for one. Area under the curve, I think, is the easiest, but if we take the value of the area under the curve, we get a function where if we look at the rate of change of that function, we get the original function back.
Rate of change of position is velocity
The integral of velocity with respect to time gives the position
Integration started to make sense to me when I started thinking about it as really fancy multiplication. For example, if you know how fast you are going and how long you have been going for, you can figure out how far you have gone. This is simple when you are going the same speed the entire time, just multiply your speed times whatever time period you care about, and boom, you have it. But, if the speed is changing, you can’t do this directly. Instead, chop up time into tiny pieces, where your speed doesn’t change much, multiply the speed then by the tiny slice of time, and add up all of the distances you get with each slice. This works for really anything that is a derivative, or something changing based on something else changing. Your speed is distance traveled by change in time. Filling a bucket with water is changing volume by change in time. You could even do something like the change in surface area over change in volume when filling a balloon. Whatever the rate, integration let’s you get the top part of that rate by multiplying it by the bottom part, when you can’t multiply directly.
Going by your analogy, go down an order, the area under a speed curve is distance traveled.
Imagine you have a curve. Now draw a big rectangle under it. The area of that rectangle is close to the area under the curve but not exactly. We can get closer by instead drawing 5 rectangles so that they are closer to the shape of the curve. If you have infinite rectangles then you will be able to add them up and get an accurate representation of the area that they fill. The math behind using an infinite number of rectangles has been simplified and named an integral.
Latest Answers