Question about integral change of variable in Infinite Server Queue example, Ross' Introduction to Proability Models

Question

Picture is from Ross' Introduction to Probability Models 11th ed.

Maybe a simple question, but I'm just wondering why the second equality in (5.17) follows. I tried substituting $y = t-s$, but that gives an extra factor of $-1$.

Answer 1

A substitution $y = t-s$ results in $\int_0^t G(t-s) ds = -\int_t^0 G(y)dy$. Notice how the limits of integration swapped. When $s$ is $0$, $y$ would be $t$ and vice versa. To put them in order, as in $\int_0^t$, you'd need to change the sign again.