Or, to really impress your teacher:
Imagine dividing the interval from a to b into n intervals and construct the Riemann sum for this function. Since each "rectangle" has has positive length base (by the construction of the Riemann sum that is always true) and positive length height (because and the height is always f(x*) for some x*), the area of each rectangle is greater than or equal to 0 and so is the Riemann sum. Since every Riemann sum is greater than or equal to 0, the integral, the limit of those Riemann sums, is greater than or equal to 0.
I have moved this thread to the Analysis subforum where, despite my signature, even the obvious requires rigorous proof.
@OP: I'm not sure what theory of integration you have studied. I suggest considering the Riemann integral in the first instance (although you might want to consider the Riemann–Stieltjes integral or even the Lebesgue integral depending on what level you're studying at).