You can’t take the two functions of the same variable separately and apply the limits separately and expect the definite integration of the product to work. For example a(y)=y² and s(y)=y³, so a(y)s(y)=y⁵. Integrated separately we have y³/3 and y⁴/4 which multiplied together are y⁷/12, not the same as y⁶/6.
But you can treat separately the sum or difference of functions, so you can have a(y)+s(y) for example. The integral of the sum is the same as the sum of the integrals.
(Don’t confuse this with multiple integrals where at least two different variables are involved.)