How about this
Table[Integrate[a[[i]] x^2, {x, 1, c[[j]]}], {i, Length@a}, {j, Length@c}]
{{215/3, 124/3, 21, 26/3, 7/3, 0}, {430/3, 248/3, 42, 52/3, 14/3, 0}, {215, 124, 63, 26, 7, 0},
{860/3, 496/3, 84, 104/3, 28/3, 0}, {1075/3, 620/3, 105, 130/3, 35/3, 0}, {430, 248, 126, 52, 14, 0}}