• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Physical interpretation of this integral equation involving distance and time?

  • Thread starter Irishdoug
  • Start date
Problem 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.
 

Ray Vickson

Science Advisor
Homework Helper
Dearly Missed
10,705
1,708
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.
 
Im asking moreso 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.
 

Ray Vickson

Science Advisor
Homework Helper
Dearly Missed
10,705
1,708
Im asking moreso 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.)
 
Ok thanks for the heads up!
 

PeroK

Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
9,072
3,277
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.
 

Want to reply to this thread?

"Physical interpretation of this integral equation involving distance and time?" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top