@Carl Love 
Thanks!
That animation is awesome.
Its confirming alreadymy former  interpretation of a scalar line integral
The vector line integral has still to be studied by me where it is standing for.

Calculs concepts can all be visualized and a proof can be followed too easier,  how it is build up
The Maple software engineers can make these topics too as animation : scalar line integral 
 

@janhardo 
Understanding a mathematical concept is one thing, but translate this into Maple is another thing.
The difficulty as i faced with deciphering the command for a PathIntegral ;)

@vv 
Thanks

Yes, a circulair disk is easier to follow and for getting more insight : a good choice.
A Curve is here a circle with length : 2*Pi*r = 2*Pi
The Pathintegral ( lateral surface of circulair disk ) = lenght circle x heigth = 2*Pi 
So the Pathintergal equals the lenght of the curve     

The spacecurve command argument handling comes in play to define all sorts of curves for a pathintegral  ( not-parametrized and parametrized)

@rlopez 

Thanks
You noticed that with all sorts of integrals involved i loose oversight. 
That a Pathintergal= Surfaceintegral (post vv 7821 )  makes that i ask myself what is that surface in your example although it is not mentioned.( answer: its a surface of a double intergral )

I tried to solve my bookexample on the same way you did, but did not get a plot yet (has to do with the definition as a spacecurve in the xy-plane)
 

@vv 

In case of the plot in the worksheet:  if i give meaning to your text then is the green area ( pathintegral) equals the red area (surface intergal) : correct?  

@vv 

Thanks
I must study my bookexample in the same manner as explained there.
Doing this in Maple on the same way.
Did not made a comparison yet for the worksheet from mr. Lopez, but it is indeed using a surface and that's not used in my bookexample

For me is interesting to see a other solution methode too , so  i am pleased to see other ones.
In my calculus book the example its solved on 3 ways! ( see pdf for one ) 
And learn from the example from mr Lopes ( handling Maple too )  
Handling the iterated integral  / PathInt  commands via context Panel Dialog is not clear for yet what to fill in, but the screenshot explains more    

Looks your formulation is the  Green integral ? what is more advanced then the basic method in my bookexample.
There is also a surface integral formula what is a iterated integral over a area in xy-plane 

 PathInt command and LineInt command  both in vectorcalculus package
PathInt ,using no vector
LineInt , is using a vector   

@rlopez 
Thanks!

Very helpful to see a graph to understand the concept of a lineintegral of a scalar
It was in the book of Calculus early trancedentials (Stewart) that i understood the concept 
(my old Dutch study book shows nothing)  
Using a Riemann sum with the rectangles and plot this as replacement for the lineintergral is a idea. 

@Kitonum 
Thanks

You can give the aera of this curvilinear integral as mass
Not work, because there is no vector involved

@Kitonum 

Thanks

It is indeed not related with vectorcalculus this example, because there is no vectorfield involved

I studied some more and i think i understand now the concept 
A visualization says more then a formula, especially in Calculus 
Maybe i can make a plot of the example 
The 2D input is not suited for a detailled plot ( a sort of fench ) for this task 
Mapleinput is needed for plotting.   

is there default command what shows vertical lines in a 2D or 3D  from domain to  functiion values ? ( must be a plot option ? )  

Thanks

The book is not helping  by explaining the method used
In Dutch language it is lijnintegraal
There are lineintegrals in Maple too( looked there), but they are all related with vectorfields, although there is a definition of curvilinear integral  under 2.a in maple help to find
But there was no example to find here.   
 

Its only by graphing this path integral for some t -values i was hoping that it becomes more clear for me

@janhardo 

I made a little start inthe vectorcalculus by drawing a 3D vectorfield. 
-don't know the type of vectorfield  and the meaning of a flowline at a given point.
I used the VectorfierldTutor() and a command generated by the tutor of the same field, but the plots differ.

Using a example in a Dutch studybook and at time there was no visulisiation of a vectorfield possible (1996/97), because no example in the book   
A 2D velocity vector field transformed as a system of a ODE and  the flowlines are integralcurves to start with. 

Download Toegepaste_wiskunde_-vectoranalyse-lijn-en_oppervlakte_integralen.mw

@rlopez 
Thanks!
Yes, Maple apps .? ..portal 
DifferentialGeometry package.has certainly my interest, but it seems not easy accesible for a starter
It's not a topic in engineering mathematics ?
Luckily these days a lot of information can be found on internet. 
Tensor  algebra/calculus as the next step after vector calculus..interesting 

@rlopez 

Thanks!

Useful to use a paper oversight what is to see in the webinar.
I looked also to : Classroom tips and techniques: vectors in the LinearAlgebra, etc 
Can't download as a Maple file on website
 

Its from 2007: does it have still importance this worksheet?  

@rlopez 
Thanks!

I studied already parts of the webinair.
It stays always difficult (impossible too) to study a particular math concept not in a classroom teached
So i must use different sources of studymaterial for vectorcalculus

Doing this with Maple is another challence.  

@tomleslie 

Thanks
Its only not the whole folium ?