@rlopez 

Thank you very much for the thoroughly answer of my question.

I do see my book handles this task (on two ways)  also with he parameter t
A line is parametrized with t and by a given line you can then come up with more then one parametrization.
But in your example is direrctly done for the curve as function y=g(x)
Manipulation on expressions is diificult to master for me, but i can learn from this example

Complete square can also of course.be used on 4*x^2 - 6*x + 4

I looked also to eleminate in the help :

The result of this elimination is an expression sequence of lists.  Each list represents a possible elimination of the given variables. => that is not the output on screen ?
A list that has sets as elements in the examples.

The result of this elimination is an expression sequenze of sets in a list as outcome in the examples for the eliminate examples in help?
Probably i am wrong and its deeper under the hood of Maple 

@janhardo 

Seems that a equation of a intersection is needed

Download vb4-blz204-toegepate_wiskunde_deel2.mw

@tomleslie 
Thanks
I was thinking on the same for this adding a t , and add more Pi  like your example
But something went wrong en got a error in the handling of the document mode without prompt
I put a restart at the top of the page and the command was working again
A restart in the middle of the document for a new calculation seems to be not correct, because then i got a error for plotting the helix.  

 

@tomleslie 

Thanks!
Just enough to do my other exercises with different curves  : these circles definitions will be combined with other plots in plot display ( if possible ) 
The vectorcalculus SpaceCurve() circle cannot be transformed into a helix 3D ?    

@tomleslie 

Thanks

-cartesian: in x and y see my example, and in parametric form   
- polar , and in parametric form.: no in3D polar becomes cylindrical
- cilindrical (your example), and in parametric form.

But not with the vectorcalculus possible ?  
A 3D positionvector with no e3 unitvector ( notation be studied in pdf mr. Lopez )

@vv 

First as PathIntegral and x^2+y^2  is a parabolide  ..all intersection planes for z > 0 shows circles   
Plotting the Curve and the scalar function on the same way like is done in the worksheet of mr Lopez
is not that easy.
- i got plot of the parabolide : x^2+y^2
-  now a circle(Curve) in the xoy plane: can be done via with (plots) or with student Vectorcalculus as position vector. ?    
- a green surface is the lift of the circle up to the surface?
 Graphical its clear what to do.

The definiton of a circle is via plots command : spacecurve  

Student vectorcalculus for SpaceCurve command has a tutor ..   

A := PathInt(x^2 + y^2, [x, y] = Circle(<0, 0>, 1));
B := PathInt(x^2 + y^2, [x, y] = Path(<cos(t), sin(t)>, t = 0 .. 2*Pi));
                           A := 2 Pi

                           B := 2 Pi

@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 ? )