The interval from 0 to 2 has been divided into eight equal intervals, starting at #0,0.25,0.5,0.75,...,1.75#. The lower Riemann sum consists of replacing the area under the curve within each interval by a a rectangle whose height is equal to the least value attained by the function in that interval. As the function #y(x) = sqrt{x}+1# is monotonically increasing, the least value of the function in each interval is attained at the left end of the interval.
Thus, the lower Riemann sum is given by
# L = 0.25 times [y(0)+y(.25)+y(.5)+y(.75)+y(1)+y(1.25)+y(1.5)+y(1.75)] = 3.685#
Similarly, the upper Riemann sum is
# L = 0.25 times [y(.25)+y(.5)+y(.75)+y(1)+y(1.25)+y(1.5)+y(1.75)+y(2)] = 4.038#
The exact value of the integral, correct to three decimal places, is
# 3.886#
