Let Q be a rectangle in En and assume that f : Q → E is integrable.
(a) Show that if f(x) ≥ 0, for all x ∈ Q, then $\int Q f ≥ 0.$
(b) Show that if f(x) > 0 for all x ∈ Q, then $\int Q f > 0.$
Solution: (a) For every partition P of Q, its lower sum L(f, P) is non-negative, so $\int Q f $ ≥ L(f, P) ≥ 0.
I have a standard solution for (b) but I do not understand why I cannot solve (b) the same way as (a) as I showed above?
You've omitted a step in the proof of (a): you must take the limit over partitions (as their mesh size tends to $0$, or however it's phrased for you). Fortunately, the limit of nonnegative numbers (when it exists, which is assumed here) is always nonnegative.
You're right that in (b), you get strict positivity for any particular partition. However, the limit of strictly positive numbers is not necessarily strictly positive. So this proof wouldn't give $\int_Q f > 0$, only $\int_Q f \ge 0$.