Integral of a position function

[motai]

What would the integral of a position function physically represent? I'm having a hard time trying to conceptualize this rather odd circumstance. I don't think it is used (at all), because usually problems deal with the derivatives of the position function and rates of change (velocity and acceleration) or using integration to find the initial position function (like parabolic trajectories) in the first place.

I'm wondering what would happen if we were to say integrate a parabolic trajectory (definite integral) and what answer would physically represent the outcome when the first fundamental theorem of calculus were to be applied.

I asked this question in my calculus class a while ago and didn't get a satisfactory answer.

Thanks

 

[Meir Achuz]

There are some mathematical results that are of no physical interest.

 

[Q_Goest]

If you integrate lengths, you get an area. If you integrate areas, you get a volume.

Why wouldn't you get a length if you integrated locations/positions?

 

[Integral]

What is your variable of integration? If you integrate with respect to time you will get a quantity with units of Length*Time. I do not recognize this as having a useful physical meaning. If you set up a path integral along the trajectory you will get the distance traveled, but this is not the same as an integral wrt time.

 

[motai]

Okay, now I see how this fits in with the Riemann Sum definition of the integral.

About the path integrals used to find distance, how is that any different from the arc length formula $$\int_a^b \sqrt{1+f'(x)^2}dx$$?

Sorry for what seems to be the silly questions... I'm just trying to push my book to the limits and questioning what the book didn't cover.

Thanks.

 

[dextercioby]

It isn't.That length arc formula is just a particular case of a first order curvilinear integral.

"Path integrals" is not a fortunate use of terms in this nonquantum case.

Daniel.