An anti-derivative is an indefinite integral. Two anti-derivatives of the same function differ only by a constant. An anti-derivative evaluated at two endpoints of an interval is the basis for the value of a definite integral. Both F(x) and F(x)-1 are anti-derivatives of the function F'(x).
A Riemman sum is an approximating sum of rectangular areas whose limiting form is a definite integral, which may have the value F(b)-F(a).