Playing around a bit.....  

If the entire problem is made symbolic with no numeric values, maple solves it into a huge equation.

But, if in the next step, this huge equation is used with all the numeric constants specified, this huge expression hangs. But after using Digits:=24 and ystart:=2000, the huge expression will look different and  the result becomes correct.

It seems impossible to make it entirely symbolic without specifying ystart. Maple gets overwhelmed.Just fascinating to test the limits. 

y2ove := y -> ap + bp*(ne*y^4 + nd*y^3 + nc*y^2 + nb*y + na) + cp*(ne*y^4 + nd*y^3 + nc*y^2 + nb*y + na)^2 + dp*(ne*y^4 + nd*y^3 + nc*y^2 + nb*y + na)^3 + ep*(ne*y^4 + nd*y^3 + nc*y^2 + nb*y + na)^4;
y2anom := x -> int(y2ove(y), y = ystart .. x)/tc;
y2anom(x);

Thank you very much !  This is awesome

A symbolic solution also suffers from the same problem and returns huge results  ( 10 ^ 11)

Is this also a result of "premature evaluation" ?  Is there a workaround  to get the correct solution ? 

I must admit that I don't understand this, but it works !!!!

plot(ta, 2000.0 .. 2050.0)

Thank you !!!!!

Thank you very much.  I agree tpn()  undefined...but when corrected -  plot still fails, 

To do something radical, I have now replaced all the functions with one single one .   y2ove(y):= a long expression

It returns the right values  y2ove(2000)  -> 12.8     y2ove(2030) -> 58.2

Integrating this function defines a new function

This also returns correct values

y2anom(2000) -> 0     y2anom(2050) -> 4.53 

So far, so good.. i guess.... but

the problem; Plotting this   should give a graph between 0 and 5 when year y goes from 2000 to 2050

plot(y2anom(y), y = 2000 .. 2050)

But the graf becomes huge (size 10^10 ).  This is strange indeed...

na := 8.069439677916595*10^5;
nb := -1.777065899098942*10^3;
nc := 1.467451715287991;
nd := -5.383733471268420*10^(-4);
ne := 7.404613067985871*10^(-8);
am := 0.77317633747818500000;
bm := -0.00626025741156560000;
cm := 0.00002185947833342660;
ap := 471.909671218139000000000000;
bp := -7.368938071612570000000000;
cp := 0.041111235018593800000000;
dp := -0.000098963929768727000000;
ep := 0.88147417256725300*10^(-7);
aq := 0.164489218115925*10^8;
bq := -32685.8887081463;
cq := 24.357113890912;
dq := -0.00806716031897729;
eq := 0.1001980162576*10^(-5);

y2ove(y) :=  ap + bp*(ne*y^4 + nd*y^3 + nc*y^2 + nb*y + na) + cp*(ne*y^4 + nd*y^3 + nc*y^2 + nb*y + na)^2 + dp*(ne*y^4 + nd*y^3 + nc*y^2 + nb*y + na)^3 + ep*(ne*y^4 + nd*y^3 + nc*y^2 + nb*y + na)^4

y2anom(x) :=  (1/700)*int(y2ove(y), y = 2000 .. x);