Creating a variable-length list of integral equations using a loop

Question

I need to automatically form a list of equations that later will be used in FindRoot. The number of equations (NN) is variable, so I need to use a For loop. When I am doing this

LstEqns = {}; For[i = 1, i <= NN, i++,
  LstEqns = 
  Append[LstEqns,
   NIntegrate[Eq[i], {x, 0, L}] == 0];
 ];

I am getting a warning that L is not a valid limit of integration (the value is not provided at this point) and in the list I am not getting i substituted by its proper values, i.e. LstEqns looks like {NIntegrate[Eq[i], {x, 0, L}] == 0,...} instead of {NIntegrate[Eq[1], {x, 0, L}] == 0,...}. The value of L is provided later (as well as actual Eq[i]), after forming the list. So the question is - how can I get a list of equations with a proper numbering? Mathematica 11.3 is what I am using if this matters.

UPD

n = 1;
LstEqns2 := 
  Inactive[NIntegrate][#[[1]], {x, 0, L}] == 0 & /@ Transpose[{eqns2}];
eqns2 = Array[Eq, n];
theEqns2 = LstEqns2;
Eq[1] = x*y + 1;
L = 1;

FindRoot[theEqns2, {y, 0}, Evaluated -> False]

• doesn't work ("The function value is not a list of numbers")

Manually copying the output of theEqns2 into FindRoot doesn't work either:

FindRoot[Inactive[NIntegrate][x*y + 1, {x, 0, L}], {y, 0}, 
 Evaluated -> False]

• same error

However, the same line as above will work if I delete inactive NIntegrate (that has a different color in Mathematica) and placeholder and then write NIntegrate again

FindRoot[NIntegrate[x*y + 1, {x, 0, L}] == 0, {y, 0}, 
 Evaluated -> False]

• this code works

Answer 1

Clear["Global`*"]

Whenever you think that you "need" a For loop, you probably don't.

LstEqns := Inactive[NIntegrate][#, {x, 0, L}] == 0 & /@ eqns

Note that what you are labeling Eq[i] are not equations but rather expressions. Since they will be the integrand and will later be the LHS of an equation, they should not have a Head of Equal

n = RandomInteger[{1, 10}]

(* 5 *)

eqns = Array[Eq, n]

(* {Eq[1], Eq[2], Eq[3], Eq[4], Eq[5]} *)

theEqns = LstEqns

EDIT: If the RHS of the equations aren't zero

n = RandomInteger[{1, 10}]

(* 6 *)

LstEqns2 := 
 Inactive[NIntegrate][#[[1]], {x, 0, L}] == #[[2]] & /@ 
  Transpose[{eqns2, rhs}]

eqns2 = Array[Eq, n];

rhs = Array[RightPart, n];

theEqns2 = LstEqns2

EDIT 2: For the revised question

n = 1;
LstEqns2 := 
  Inactive[NIntegrate][#[[1]], {x, 0, L}] == 0 & /@ Transpose[{eqns2}];
eqns2 = Array[Eq, n];
theEqns2 = LstEqns2;
Eq[1] = x*y + 1;
L = 1;

theEqns2 must be activated; and since FindRoot has the attribute HoldAll, you must Evaluate the first argument, i.e.,

FindRoot[Evaluate[theEqns2 // Activate], {y, 0}, Evaluated -> False] // Quiet

(* {y -> -2.} *)