Yes it does, but that's exactly what you have to show. How you gonna do it?
Let f be an integrable function on [a,b]. Suppose that f(x)≥0 for all x and there is at least one point ∈ [a,b] at which f is continuous and strictly positive. Show that
b
f(x)dx>0
a
I think I am missing something because it looks too simple, since the function is continuous and strictly positive doesn't the integral have to be >0?
Note, however, that the problem does NOT say that f is continous on [a, b], only that it is continuous at . But you can show from that there exist some interval around in which f is continous and positive. The integral over that interval is positive and f is never negative.