Thanks, looks good.
Try to add some more text to get even a clearer picture.
This procedure is a nice basis to extend to more complicated surfaces f(x,y,z) with a spatial domain curve.
This example was already very instructive.

Thanks, Admit it may be a bit unclear, will try to word it more clearly.

Integral expression = Elliptic expression = float expression 

That can be shown in the plot itself this above output.
Also outside the plot would be useful to show the outcome of the procedure.  

Thanks , It seems that in the procedure there is a provision if a special function occurs in the arc length ( red curve) calculation then a numeric value of this is shown?
Can the special function symbolic expression  just remain with the numeric value behind it in the plot?
 

VisualizeFunctionAndCurve(2*x+y^2, sin(t)+1, 2*cos(t), 0, [0, 1/2*Pi]);

Thanks, yes this definitely became an overhaul of the procedure ( I didn't expect )
I definitely need to study this : how some things have been changed and why.
What is the objection to using a function operator instead of an expression notation ?
This procedure is still for simple surfaces it appears
For more complicated surfaces it becomes f(x,y,z) and the domain becomes a space curve
I still want to try to extend the existing procedure... 

If this should work the VisualizeFunctionAndCurve () procedure, then a further modification for the domain curve  

A plane curve can be represented in two ways: by an equation H(x,y)=0 (the implicit description) and parametrically by a pair of equations with one parameter (the parameter description).

@sija 
Thanks, i will look at it again

Interesting, but from the looks of it, none of the animations work in Maple 2024, or am I mistaken? 

Have revisited the task and have the idea that this elaboration could be better in Maple ? 

Download lijnintegraal_2e_poging_maple_primes.mw

@Carl Love 
I asked the AI for an alternative to the is () command and it didn't come up with a thoughtful answer anyway.
Fine to is ( ) command and apparently there are no alternatives to this?
I find it tricky with that evaluation in Maple with the cauchy-riemann equations.

@janhardo 
Can I also do that check on the cauchy riemann equations via : evalb ?

if evalb(u_x = v_y) and evalb(u_y = -v_x) then
    printf("The function is analytic (holomorphic) at this point.\n");
    printf("The derivative f'(z) is %a + I*%a\n", u_x, v_y);
else
    printf("The function does not satisfy the Cauchy-Riemann equations and is not analytic.\n");
end if

@Carl Love 
Thanks, Yes, that "is" command is essential to establish that the cauchy-riemannt equations are true for all variables

@Axel Vogt 
Thanks, forms a nice basis to rewrite the procedure and the hereby use the D operator instead of diff command.

@acer 
Yes, when to use a branch from a topic?
There are several ways to derive that zeta(2) value. 
I thought this was 1 branch to explore this further and another branch for a different approach to derive zeta( 2).
Otherwise it will be one long thread of proof methods topics , fine by me too and forget the branches.

How about checking this with ODESteps in Student package ?

@vv 

Thanks, I think you are absolutely right to study the methods of evidence.
The difficulty for me is still how to start translating this into Maple, and your elaboration is an example of how it can be done.