resultant makes sense only with respect to a single variable. So, resultant(f, g, x) eliminates (roughly) x for the system {f=0, g=0}.
If you want to eliminate both x and y, use eliminate or apply resultant twice.
Actually, you should say what exaclty you want to achieve.

int(simplify(v), x);

works.

There is a bug in Maple 2020 which does not integrate the obvious integral

int(sqrt(-x*(x - 20))*sqrt(-1/(x*(x - 20))), x = 0 .. h) assuming h>=10,h<=20;

( should be of course h) .

Workaround:

f := sqrt(-x^2+20*x):
F := simplify(f*sqrt(1+diff(f,x)^2)) assuming x>0, x<20:
solve(2*Pi*int(F, x=0..h) = 1005);

          201/(4*Pi)

It is easy to see that if there was a solution, then 
s  :=  a^4 + 4*a^3*k + 2*a^2*k^2 + 4*a*k^3 + k^4
would be an exact square for a,k positive integers (actually 0 < k < a).

But s is never a square for 0 < k < a. This can be proved mathematically, or checked by Maple for a <= N = 5000 say:

N:=5000;
seq(seq(`if`(issqr(a^4 + 4*a^3*k + 2*a^2*k^2 + 4*a*k^3 + k^4), [a,k],NULL), k=1..a-1),a=1..N);

You define u(x,t) := ...,  but u is already defined as a table u[i]:=...
This generates a total mess. Use another "variable", e.g. U(x,t) instead.

How can you believe that the expressions could be equal? Not only the exponents of sin(...) are different (2/n  versus 2/(n-1)) but the constants are not the same, one of them is piecewise, ...

C:=(m::nonnegint,n::integer) -> 
  coeff(coeff(product((1-q^i)*(1-q^i/z)*(1-q^(i-1)*z), i=1..m+1),q,m),z,n):

C(6,2),  C(6,4); 

      0,  1

(k is an arbitrary integer)

restart;
f:=(x,c,d)->(x-c)*(x+c)*(x-d)*(x+d):
g:=(x,c,d)->(x-c)*(x+c-1)*(x-d)*(x+d-1):
N:=10:
for c to N do  for d from c+1 to N do
  for x1 to c-1 do
    for F in [f,g] do
      if nops(factor(F(x,c,d)-F(x1,c,d)))=4 then lprint(F(x+k,c,d),m=F(x1,c,d)) fi;
    od
  od
od od:

(x+k-4)*(x+k+3)*(x+k-5)*(x+k+4), m = 180
(x+k-4)*(x+k+3)*(x+k-6)*(x+k+5), m = 360
(x+k-4)*(x+k+4)*(x+k-7)*(x+k+7), m = 720
(x+k-5)*(x+k+4)*(x+k-7)*(x+k+6), m = 504
(x+k-5)*(x+k+4)*(x+k-9)*(x+k+8), m = 1260
(x+k-5)*(x+k+4)*(x+k-10)*(x+k+9), m = 1800
(x+k-5)*(x+k+5)*(x+k-10)*(x+k+10), m = 2016
(x+k-6)*(x+k+5)*(x+k-7)*(x+k+6), m = 1260
(x+k-6)*(x+k+6)*(x+k-7)*(x+k+7), m = 1440
(x+k-6)*(x+k+5)*(x+k-9)*(x+k+8), m = 1080
(x+k-7)*(x+k+7)*(x+k-9)*(x+k+9), m = 2880
(x+k-7)*(x+k+6)*(x+k-10)*(x+k+9), m = 3780
(x+k-8)*(x+k+8)*(x+k-9)*(x+k+9), m = 5040
(x+k-9)*(x+k+8)*(x+k-10)*(x+k+9), m = 5544
(x+k-9)*(x+k+8)*(x+k-10)*(x+k+9), m = 2520
 

1. In Maple 2020 it can be done because a (statement)  is an expression

restart;
x:=10: str:="A":
str:=cat(str,  `if`(x=10,  [(x:=11)," it was 10"][2], [(x:=8)," it was not 10"][2]));  x;

2. `if`  is alias for ifelse  and a link appears in the help page of if. Or, use ?ifelse 

p:=randpoly([x,y,z]);
`%+`(sort([op(p)], key=abs@coeffs)[]);

Use value(%)  to  go back.

This isthe well known  Pell's equation.

S:=isolve(61*x^2+1=y^2):
for _Z1 from 0 to 6 do
lprint(eval([x,y],rationalize(S[4]))[])
od;

0, 1
226153980, 1766319049
798920165762330040, 6239765965720528801
2822295814832482312327709940, 22042834973108102061352541449
9970149719303180503641083029374964080, 77869358613928486808166555366140995201
35220930741174421456911021812718768924061809900, 275084262906388245923976756042747916825335226249
124422801783292138491822391332416163557158135530198606120, 971773147303355325052564141449134520779147876502526039201
 

It is easy to obtain such equalities. Here is a simple example. It can be extended to a generic one.
Why do you consider them so important?

g:=proc(z)
   local m:=subs(sqrt(3)=0,z),
         n:=z-m;
   m^2+n^2-1+ 2*m*n-sqrt(3)
end: 
F:=c->sqrt(1+sqrt(3)+c):
G:= n -> (F@@n)(g@@n)(1+sqrt(3))=1+sqrt(3):
G(3);G(4);G(5);

You do not have a value (true or false) for print_table
Note also that a better  header for your proc is:

proc(f::procedure, a::realcons, b::realcons, N::posint, print_table::truefalse)
 

You must use parallel substitutions in ex:
ex := simplify(subs([ X = cos(theta)*X-sin(theta)*Y, Y = sin(theta)*X+cos(theta)*Y ], eq)); 

(You have defined the procedure f. It is then easier to use f(X-9/7, Y+8/7)  and similar for ex, instead of subs) .

It is interesting that for diff, the first argument can be almost anything, including a mathematical nonsense. For sets, lists, rtables it acts as expected (elementwise), but even strings are accepted. 
Note that series may produce errors.

f:=sin(x) + x^3*"a";
g:=sin(x) + [cos(x), x^3];
h:=sin(x) + [cos(x)*"a"] + {x^2*"b"^3 + "c"};