Calculation
Of R_{n}[1, 3]

Fig. 2.1
Finding Areas. 
See Fig. 2.1. Use a regular partition of order n of [1, 3] and the left endpoints. The
length of each subinterval is
(3 – 1)/n = 2/n. The left endpoints are:
_{}
Calculation
Of R_{n}[0, 3]
Refer to Fig. 2.1.
Utilize a regular partition of order n of [0, 3] and the left endpoints. The
length of each subinterval is
(3 – 0)/n = 3/n. The left endpoints
are x_{i} = 0 + i(3/n) = i(3/n)
for i
= 0, 1, 2, ..., n – 1, so they are:

Fig. 3.4
Upper Riemann Sum.

See Fig. 3.4. Recall
that the endpoints are: