Using already found functions  f ,  y  we form 3 lists (they are the columns of your table) and plot them in the form of broken lines:

Exact:=[seq([k,f(k)], k=0..1, 1/4)];
Numeric:=[seq([k,y(k)], k=0..1, 1/4)];
Error:=[seq([k,f(k)-y(k)], k=0..1, 1/4)];
plot([Exact, Numeric, Error], color=[green,blue,red], legend=["Exact", "Numeric", "Error"], axes=box);
                                  

We see that the blue and green lines between points 0.25 and 0.75 almost coincide.

restart;
T:=1:  K:=20:
f:=x->piecewise(x>=-1/2 and x<=1/2,1, 0):
eq:={diff(x(t),t)-add(f(t-k*T), k=0..K)*x(t)=0, x(0)=0}:
sol:=dsolve(eq, numeric);
 

Here is another solution in which the lifting to the mountain occurs at a constant rate of 1. The differential equations are solved numerically, because there are difficulties with the symbolic solution. Here, we plot the motion trajectory in 3D and do the motion animation in 3D:

Download path1.mw

Your error is that you consider the gradient as a constant. In fact, it depends on the point  (x,y)  and to find the way to the top of the mountain differential equation is required:

Download path.mw

If you need to write this as an inert sum (not for calculation, but for example for a presentation), then you can do it like this:

restart;
2^(k+1)*Sum(sqrt((2*r-1)*(2*s-1))*psi[n*(M+1)+s](t), s=`and`(1,(s+r)::odd) .. r-1) ;

I can not confirm your claim. In Maple 2018 we have:

0^0;
limit(x^x, x=0);
plot(x^x, x=0..2, 0..4);
                                        

PS. I just noticed that you have a very old version of Maple (20 years ago). Since then, this error has long been corrected.

You can use inert sqrt  %sqrt  or  ``(...)  construction:

ex:=sqrt(x)/2/h;
ex1:=%sqrt(ex^2);  # Or
ex2:=sqrt(``(ex^2));

#  Examples of use
simplify(ex1) assuming h>0;  # Or
value(ex1) assuming h>0;  # Or
simplify(expand(ex2)) assuming h>0;
 

A way:

restart;
assign(seq(seq(m[i,j]=x^i*y^j, j=1..5), i=1..5)):
w:=m[2,3]+m[3,2];
v:=m[1,5]-3*m[2,2]+2*m[3,4];

Output:                       
                                 
                                

Download cp.mw

Yes, we can plot the entire graph without using differential equations according to the following plan:

1. First, we plot the graphs of  f , g , h of the implicit functions, from which we find the maximum range for  t - variable, so that there are real values of  x , y , z.

2. Then, with step = 0.01, we solve the equations for these variables. For every  t  we get  2 values of  x , 2 values of  y  and 1 value of  z .

3. Thus we get 4 branches of the entire graph and plot them.

The final result:

All the codes in the attached file.

curves_4.mw

x , y, z  variables are specified by the equations f , g , h  implicitly. These equations have real solutions not for every  t = 0 .. 2*Pi. This is clearly seen from the graphs of implicit functions. For example, from the graph  f , we see that for  t>3.48...  and  t<2.2...  there are no real solutions:

plots:-implicitplot(f, t=0..2*Pi, x=-10..10, gridrefine=5, view=[0..2*Pi, -1.5..1.5]);
                         

Edit.

You can get an symbolic approximate value of this integral by expanding the integrand in a series in powers of  x :

restart;
S:=12*x^3*c[2]+6*x^2*c[1]+x^2*exp(x^3*c[2])*exp(x^2*c[1]):
P:=convert(series(S, x=0, 10), polynom);
int(P, x = 0..1);
                             

In  s:=LowerCase(s):  line of your procedure there are 2 errors at once:

1. You can not assign anything to formal parameters in the body of a procedure.

2. Strings can not act as names.
 

Here are 2 examples:
It does not work:
P:=proc(x::string)
uses StringTools;
x:=UpperCase(x);
%;
end:
P("y");

But it works:
P1:=proc(x::string)
local s;
uses StringTools;
s:=UpperCase(x);
s;
end:
P1("y");

Edit.
 

f:=x->x^2*sqrt(25-x^2):
a:=0.: b:=5.: n:=100: h:=(b-a)/n: S:=0:
for j from 0 to n-1 do
S:=S+(f(a+j*h)+f(a+(j+1)*h))/2*h;
end do:
S;

restart;
with(plots):
local gamma:
g:=2:
rr:=3:
V1:=arrow([0.5,0], [-4.5,0], width=[0.006, relative=false], head_width=[0.07, relative=false], head_length=[0.2, relative=false], view=[-4..4,-1..1]):
V2:=arrow([1.5,0],[2.5,0], width=[0.006, relative=false], head_width=[0.07, relative=false], head_length=[0.2, relative=false], view=[-4..4,-1..1]):
Seg:=plot(0, x=1/g+0.1..rr/g-0.1, color=black, thickness=7):
C:=plot([[0.55,0],[1.45,0]], style=point, color=black, symbol=circle, symbolsize=14):
L:=plot([[0,t,t=-0.04..0.04],[1,t,t=-0.04..0.04],[2,t,t=-0.04..0.04]], color=black, thickness=5):
T:=textplot([[0,-0.15,0],[1,-0.15,1],[0.55,-0.15,1/gamma],[1.45,-0.15,r/gamma],[2,-0.15,r]], font=[times,bold,16]):
display(V1,V2,Seg, C, L, T, size=[800,400], axes=none);

   Output: