In[3]:= With[{a = 1.5},
Integrate[Integrate[x^5 Exp[y^2], {y, x^2, a^2}], {x, 0, a}]]
Out[3]= 53.5678
Or, via a general symbolic solution:
In[5]:= Integrate[Integrate[x^5 Exp[y^2], {y, x^2, a^2}], {x, 0, a}]
% /. a -> 1.5
Out[5]= 1/12 (1 + (-1 + a^4) E^a^4)
Out[6]= 53.5678
