Let f(x)= (x-(1/2))^2 + 3, 0<= x <= 1. If the interval [0,1] is partitioned into 4 subintervals of equal length, then what is the value of the smallest Riemann sum for f(x) and this partition?
they want you to do the lower Riemann sum
so you would partition the interval into [0,1/4], [1/4,1/2], [1/2, 3/4], and [3/4,1]
in each of these intervals, find the point that makes $\displaystyle f(x)$ the smallest, and then use those points as your $\displaystyle x_i$'s in the formula
$\displaystyle \int_0^1 f(x) \approx \sum_{i = 1}^{4} f(x_i) \Delta x$
where, of course, $\displaystyle \Delta x = \frac {1 - 0}4 = \frac 14$
I wound up getting $\displaystyle
\sum_{i = 1}^{4} \frac{11}{16}
$ for the first interval,
$\displaystyle
\sum_{i = 1}^{4} \frac{47}{64}
$ for the second interval,
$\displaystyle
\sum_{i = 1}^{4} \frac{3}{4}
$ for the third interval, and
$\displaystyle
\sum_{i = 1}^{4} \frac{49}{64}
$ for the fourth interval,
I'm not sure what to do from there?