Physical interpretation of this integral equation involving distance and time?

In summary, the conversation discusses the interpretation of an integral in relation to position and time. The conversation also highlights the importance of understanding that not every equation needs a physical interpretation and that it is more important to focus on the mathematical principles rather than constantly trying to give a physical meaning to everything.
  • #1
Irishdoug
102
16
Homework Statement
I have been given a generic question, however I don't know how to interpret it if the variables had an assigned meaning.
Relevant Equations
f'(x) = ##\frac{d}{dx}## (##\int## ##\frac{1-t^2}{1+t^2}## *dt). The integral has the limits 0-x.
I am able to solve the problem however if x was position and t was time how is this problem interpreted?

I know, for example that ##\frac{dx}{dt}## tells us how the position of something changes as time changes (or instantaneous change) and an integral gives a net change so to speak.
 
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  • #2
Irishdoug said:
I am able to solve the problem however if x was position and t was time how is this problem interpreted?

I know, for example that ##\frac{dx}{dt}## tells us how the position of something changes as time changes (or instantaneous change) and an integral gives a net change so to speak.

You can easily write ##\int_0^x (1-t^2)/(1+t^2) \, dt##; just right-click on the image to see the LaTeX commands.

Anyway, why should ##f(x)## and/or ##f'(x)## have any particular "interpretation"? Maybe this is just a problem in calculus without any interpretation or without any relation to distance and time.
 
  • #3
Im asking more so about the RHS of the equation. The reason I want to know how to interpret it as I do physics and equations require a physical interpretation.
 
  • #4
Irishdoug said:
Im asking more so about the RHS of the equation. The reason I want to know how to interpret it as I do physics and equations require a physical interpretation.

If the integral arises in some "application" in physics, engineering, chemistry, biology, economics, ... then you may be able to come up with a reasonable interpretation. If it is just a calculus practice-example there may not be any interpretation, and you are wasting your time looking for one. It would be much, much better to just realize that calculus is a branch of mathematics that is widely applicable to numerous fields and situations, and that not everything need be related to physics. Equations definitely do NOT need a physical interpretation, although you may find it difficult to accept this. Basically, I am saying that you are hindering your own learning process by constantly trying to give a physical interpretation to everything. (I say this as somebody who earned a PhD in physics from MIT.)
 
  • #5
Ok thanks for the heads up!
 
  • #6
Irishdoug said:
I am able to solve the problem however if x was position and t was time how is this problem interpreted?

I know, for example that ##\frac{dx}{dt}## tells us how the position of something changes as time changes (or instantaneous change) and an integral gives a net change so to speak.

##t## looks more like a dummy integration variable here than time. For example, you can (almost) write:

##f(x) = \int_0^x g(x) dx##

But, technically, you are using ##x## as both the dummy integration variable and the variable for which your function is defined.

In general it's better to use different symbols for the two variables. For example:

##f(x) = \int_0^x g(x') dx' = \int_0^x g(u) du##

Or, in your case, ##t## is used as the dummy variable.
 
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1. What is the physical meaning of the integral equation involving distance and time?

The integral equation involving distance and time is a mathematical representation of the relationship between an object's position and the time it takes to travel that distance. It can be used to calculate the displacement, velocity, and acceleration of an object over a given time interval.

2. How is the integral equation involving distance and time used in real-world applications?

The integral equation involving distance and time is used in many real-world applications, such as in physics, engineering, and economics. It can be used to model the motion of objects, calculate the work done by a force, and determine the area under a curve in economic analysis.

3. What are the units of the integral equation involving distance and time?

The units of the integral equation involving distance and time depend on the specific application and the units used to measure distance and time. In general, the units will be a combination of distance units (such as meters or feet) and time units (such as seconds or hours).

4. How is the integral equation involving distance and time related to the concept of displacement?

The integral equation involving distance and time is directly related to the concept of displacement, which is the change in an object's position from its starting point. The integral of velocity over a given time interval gives the displacement of an object during that time.

5. Can the integral equation involving distance and time be used for non-uniform motion?

Yes, the integral equation involving distance and time can be used for both uniform and non-uniform motion. In the case of non-uniform motion, the velocity function will be a function of time, and the integral will need to be evaluated using techniques such as integration by substitution or integration by parts.

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