4. Let f(x) = 2x + 1 for 0 £ x £ 1. If the interval [0,1] is partitioned into 4 subintervals of equal length, then what is the smallest Riemann sum for f(x) and this partition?

The Riemann sum is an approximation to the definite integral and is given by

Where, n is the number of subintervals and x_i^* is any point within the ith subinterval and D x_I is the length of each ith subinterval. Since the interval [0,1] is partitioned into 4 subintervals of equal lengths then the set of partition points are,

where,

Since, the given function is strictly increasing, then choosing the left end point of each partition will result in the smallest Riemann sum approximation. Thus, choosing

we have the following sum,