@Pepini You have done a good job in following the outlined steps.  To get the correct curve, you need to change the line

N1 := simplify(N/Norm(N))

to

N1 := simplify(N/Norm(N,2))

See the documentation of Norm for explanation.

@Pepini How did you get the equation of the green curve?  You have plotted it in the range t=0..Pi/4.  If you plot it in the range t=0..2*Pi you will see that it is far from being a great circle.

Here is a sketch of how to obtain the correct equation of the green curve, assuming that you want it to be an arc of a great circle.

Let A and B be the points of intersection of the green curve with the blue and red curves, respectively.  The Cartesian coordinates of those points are known.  Consider A and B as the endpoints of unit vectors emanating from the origin.  The green curve lies in the plane formed by those two vectors.  Call that plane P.

The vector N = A x B (the cross product) is perpendicular to the plane P.  Normalize N by dividing it by its own length.

Let M = N x A.  Then M is a unit vector that lies within the plane P and is perpendicular to A.  Thus, the pair (A, M) forms an orthonormal basis within the plane P.  The parametric equation of the great circle corresponding to the green curve is C(t) = A*cos(t) + M*sin(t).

We have C(0) = A.  We want to find T so that C(T) = B.  We note that C(T) = B is three equations in one unknown, T.  So any of the three equations may be used to solve for T.  The middle equation is the simplest.  Solving it we obtain 

T:= arctan(sqrt(2)*sqrt(13 - 2*sqrt(2))/(2 + sqrt(2)))

Finally, we perform the integration that you have shown in your original question, and arrive at the expression for the curve length that I showed in my previous reply.

@acer Thanks for looking into this.  As you and mmcdara have noted, this used to work properly in earlier versions of Maple.  I will report the issue to Maple Support.

PS: Oops, I just noticed that you said that you will file a report.  I won't submit a report then.

@mmcdara Thanks for verifying the issue and especially pointing out that this used to work correctly in earlier versions.  I will file a report with Maple Support.

@Kitonum Thanks for confirming the issue and supplying a workaround.  As mmcdara has noted, this is indeed a bug.  I will let Maple Support know.

@nm You don't want u(0,theta)=0.  You want u(0,theta) to be finite.  So we go back to the solution that you have obtained

and apply expand() to get

We see that the last term blows up at the origin. To remove that signularity, we let _C2 = 0:
factor(eval(sol, _C2=0));

which is the correct solution.

@vv Those are good observations.  I wonder how Maple arrives at zeros for the values of those integrals.

@mmcdara Thanks for the idea. This has helped me a lot since I have about 200+ such integrals to calculate.

@Kitonum Thanks for the solution. Actually I thought of the same thing yesterday after I had posted the question.   Mmcdara's answer is more suitable to my needs, but in the absence of that, your solution would have been also usable.

@Thomas Richard As you and Preben have noted, there were a few careless errors in the original worksheet. I have replaced it with a corrected one.

I wouldn't have noticed the typo in theta vs thata since as it happens, the thata was in a conditional branch which is not reached. Thanks to you, and to maplemint, for pointing it out.

As to uses plots, plotttools, I haven't encountered issues in the past by loading plots and plottools globally and then calling their comnands from within procs. I assume the only drawback of doing it my way is that those procs are no longer self-contained / portable, but that's not a concern in this one-off application. If there are other adverse consequences, I would be interested to know.

@Preben Alsholm You are correct in both spotting the errors, and your remedies.

I made a few last minute changes to the worksheet before posting and carelessly did not verify their correctness.  I have uploaded a corrected version now.

@Carl Love Thanks for the compliments. As you have guessed, this worksheet does not present notable technical difficulties.  The one aspect that offered a bit of challenge was that the wheel's spinning is not a simple rigid rotation; its parts move relative to each other as it spins.

@Thomas Richard I didn't know about Maple's intsolve. That's very useful to know.  Thanks!

However, applying intsolve to the integral equation that I showed earlier, says::
Error, (in intsolve) integral equation is not linear.
That's rather odd, because the equation is obviously linear.

It will be good if you could bring this to the attention of the developers.

@Realeboha Maple's "diff" operator takes the traditional derivatives of integer orders. For fractional order derivatives use "fracdiff", as in

alpha := 1/2;
fracdiff(g(t), t, alpha) = g(t)*t^(1-alpha)/GAMMA(2-alpha);

This results in the integral equation

From here on you are on your own since Maple has no facilities for solving integral equations.
 

@acer When you file this, you may want to increase the font size to 64 to make it very obvious that space is being allocated for labels even if they are empty strings.