The blue line in the graph is part of the x-axis.
Note. The definition of Riemann Sums will be given in the development that follows.
which is called a partition of [a,b].
If each f(xi) > 0 then the area of the ith rectangle is
and the sum of the areas of the rectangles is then:
More generally, we do not require that f(xi) > 0 as we define
If you want to view some additional graphs illustrating Riemann Sums with different values of n and different choices of xi's, then make your choices from the following two groups of options:
Note that the Riemann sum when each xi is the right-hand endpoint of the subinterval [ai-1, ai] is
when each xi is the left-hand endpoint of the subinterval [ai-1, ai] is
and when each xi is the left-hand midpoint of the subinterval [ai-1, ai] is
Summary of the material above.