# Understanding Calculus in a different way

First, here's a question.
One can integrate the equation for the volume of a sphere, but does it have any real world meaning? When I think of integration, I think of "Area under the curve integrated". But what does the integration of the eq for volume mean? Density? Or nothing at all. Is there a branch of investigation concerning the meanings of derivation and integration because it's pretty interesting to try to understand it in terms of physical meanings.
I know that there is meaning for the rate of change of velocity and so on but it's not readily intuitive to most. So maybe there's a meaning to the area under the curve 4/3 pi R ^3.
That equation can be plotted so there must be an area under it above the x axis. So what does this area imply?
I've search hi and lo on the net and have never seen any discussion on this. If one can develop an intuitive sense of calculus, it would be an easier topic to take on.
Thx,
Rocky

HallsofIvy
Homework Helper
I'm not sure why you think "integral of the volume of a sphere", by which I guess you mean $\int (4/3)\pi r^3 dr$, should have any "real world meaning". Mathematics, in general, doesn't have "real world meaning" until you assign a meaning to the basic concepts.

:) So are you saying that someone created the concept of volume by integrating the eq for the surface area of a sphere?? I don't know if I go along with that. I think the concept of volume had been around long before calculus. So someone found out that the volume of a sphere could now be determined by manipulating the eq for the surface area of sphere.
Otherwise, I'd make up some geometrical object that corresponds to the integral of the volume of a sphere and call it a "rocky".
I'm probably not making myself understood but I think it's perfect logical to assume that the integral of the vol of a sphere means something in the real world, just as the derivative of acceleration means something.
Yes?

matt grime
Homework Helper
I'm probably not making myself understood but I think it's perfect logical to assume that the integral of the vol of a sphere means something in the real world, just as the derivative of acceleration means something.
Yes?
No. Not really. There is, I'm sure, some mileage to be had in a philosphical debate about 'meansing' in the 'real world'. But I see nothing logical in supposing that a completely unrelated concept must imply something about spheres and their volumes.

A better analogy would have been to assert that the integral of acceleration means something. Does that throw it into perspective? You can define it; is it useful? Does your integral of the volume function do anything useful? It gives a function whose derivative is the volume function. I'm not sure that this is useful, but it may be in a physical question where we may have a something that depends on volume (or mass), and where the integral is useful.

Maybe I SHOULD be using derivation and not integration...

* Velocity is the rate of change of position.

* Acceleration is the rate of change of velocity

One might just stop there and say there is no real world quantity if you keep applying the derivative. But..

* Jerk is how fast something accelerates with respect to time.

There are several more physical quantities that are revealed upon further derivation beyond Jerk.

One may never have assumed this. Therefore I am extrapolating in the other direction that as you continually integrate a geometrical concept, you will get new geometrical entities that have been conceived of or possibly not conceived of.

In physics, jounce or snap is the fourth derivative of the position vector with respect to time, with the first, second, and third derivatives being velocity, acceleration, and jerk, respectively; in other words, the jounce is the rate of change of the jerk with respect to time.

Currently, there are no well-accepted designations for the derivatives of jounce. The fifth and sixth derivatives of position as a function of time are "sometimes somewhat facetiously" [1][2] referred to (in association with "Snap") as "Crackle" and "Pop", from the cereal characters; however, these terms have not gained widespread acceptance.

As one can see here, further derivations of the position vector lead to at the very least fascetious designation of quantities.

Is there something unique about position that teases one to do this iterative derivation. Why is iterative integration meaningless beyond the volume of a sphere?

In the context you are looking for, it may very well be meaningless. Just because you can do something doesn't make it useful. Or it could be that we haven't found a useful thing that it models yet.

Integration and differentiation are just operations that can be performed on formula. They don't have any real-world meaning intrinsic to them, but the operations they perform can have very useful real world meanings when performed on the right kind of formula (calculating velocities, volumes, etc). So it's perfectly possible to apply the operations to a vast array of formula, but just because some applications model something in the real world doesn't mean that all of them do (or that we have discovered all the useful applications)

Thanks.
So I guess SNAP (rate of change of Jerk), CRACKLE, and POP have no real world meaning as well as it relates interative differentiations of distance.

Why is it that the rate of change of acceleration leads to Jerk instead of nothing meaningful?
Just luck?

What makes the difference?

Why is integration of circle to surf area to volume, doomed to end there and not continue beyond volume? I would ask this on a philosophy forum but I am actually looking for a real answer.

Hmm. I'm wondering...in each case we are adding a dimension. So maybe the integral of the volume of a sphere has no real world value because we only know the world in three dimensions and the result is R to the fourth power.

Am I on to something here?

It may be it has a meaning we haven't discovered... it may be there is no reason. I haven't heard of a meaning for the integration of volume though.

I have discovered a relationship. If you integrate the equation for the volume of a sphere, you get an equation that is a cone whose height is the radius squared, such that as the radius increase the height increased as the square. Now, my next question is this a new geometrical object? Does it have an analogue in nature? Is it just a specific case of a generalized formula since you could have an infinite variations on the power that R is raised to? Why does integrating the volume of a sphere eq, give this. Is there something to be learned from it. Or just a coincidence that it fit any geometric specifici case?
At least I'm making progress. :)

Your cone is a three dimensional object so the volume is measured in unit^3. You can always force some description to justify the cone interpretation.

But, I think a more natural way to picture this integral is as follows:

Imagine a point in space and time (x,y,z,t) starting at (0,0,0,0). Then, say that a sphere starts to grow centered at that point (0,0,0) in space at time t=0. Assume the growth is linear with respect to time. For example, let r(t)=t [radius at time t (in sec) is t (meters).] So, after 1 sec, the radius is 1 meter. After 2 sec, the radius is 2 meters. Now, we can integrate the volume of the sphere over time for t=0 to t=R.

So, integrate (4pi/3)t^3 from t=0 to t=R. You get (pi/3)R^4.
The units are (meters^3)*(sec). This can be viewed as the space-time “volume” of the growing sphere for time t=0 to t=R. It is the amount of space-time contained in the growing sphere.

If the radius of the sphere grows at the speed of light, c, then use r(t)=ct to get:

c(pi/3)R^4, the space-time “volume” contained in the “light cone” (per special relativity.) But “cone” here is derived from the 2 dimensional space version plus one dimension for time, which yields a 3 dimensional cone.

In your example, we could call it a 4 dimensional cone, whatever that would look like.

First, here's a question.
One can integrate the equation for the volume of a sphere, but does it have any real world meaning? When I think of integration, I think of "Area under the curve integrated". But what does the integration of the eq for volume mean? Density? Or nothing at all. Is there a branch of investigation concerning the meanings of derivation and integration because it's pretty interesting to try to understand it in terms of physical meanings.
I know that there is meaning for the rate of change of velocity and so on but it's not readily intuitive to most. So maybe there's a meaning to the area under the curve 4/3 pi R ^3.
That equation can be plotted so there must be an area under it above the x axis. So what does this area imply?
I've search hi and lo on the net and have never seen any discussion on this. If one can develop an intuitive sense of calculus, it would be an easier topic to take on.
Thx,
Rocky
the integral is a sum over a function (like area under the curve). It has many applications - i.e. the integral of the velocity is the position, and in more advanced fields, such as electricity and magnetism, you can integrate a potential field over a line to get the potential difference, find the amount of work needed, etc... its simply another tool, this time calculating a sum - the applications vary widely