you can check the same wikipedia article i suggested in your last post to see why it is defined that way
there are several ways to get to that, or a similar expression. none of these ways are really short though, as long as we are integrating by definition (i'm talking about things like Simpson's rule, the midpoint rule, ...). the shortcut would be the regular integral that you may have seen before, which isAnd the second question: Do we always have to use a Riemann sum to arrive at this part: ? What, no shortcut?