how to solve limited integration

Question

how to solve integration  from (400 to 800) of this equation {(a(y)s(y)dy) over integration  from (400 t0 800) s(y)dy }.

when i have the value of (a and s) for each  point from (400 to 800).

can I just put the value of a+s/s for each point. I mean each value of a & s from 400-800.

Answer 1

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.)