Define $$x = \sum _ { i = 1 } ^ { 10 } \frac { 1 } { 10 \sqrt { 3 } } \frac { 1 } { 1 + \left( \frac { i } { 10 \sqrt { 3 } } \right) ^ { 2 } } \quad \text { and } \quad y = \sum _ { i = 0 } ^ { 9 } \frac { 1 } { 10 \sqrt { 3 } } \frac { 1 } { 1 + \left( \frac { i } { 10 \sqrt { 3 } } \right) ^ { 2 } } .$$ Show that a) $x < \frac { \pi } { 6 } < y$ and b) $\frac { x + y } { 2 } < \frac { \pi } { 6 }$. (Hint: Relate these sums to an integral.)

$\lim_{n \rightarrow \infty} \frac{3}{n}\left\{4 + \left(2 + \frac{1}{n}\right)^{2} + \left(2 + \frac{2}{n}\right)^{2} + \ldots + \left(3 - \frac{1}{n}\right)^{2}\right\}$ is equal to
(1) 12
(2) $\frac{19}{3}$
(3) 0
(4) 19

$$\begin{aligned} & f : [ 1,3 ] \rightarrow [ 2,10 ] \\ & f ( x ) = 1 + x ^ { 2 } \end{aligned}$$
The interval $[ 1,3 ]$ is divided into two subintervals of equal length, and the right endpoints of these subintervals are marked as $x _ { 1 }$ and $x _ { 2 }$. Then, two rectangles are drawn with each subinterval as the base and heights $f \left( x _ { 1 } \right), f \left( x _ { 2 } \right)$ respectively.
If the sum of the areas of these rectangles is A and the area of the region between the function f and the x-axis is B, what is the difference A - B in square units?
A) $\frac { 11 } { 2 }$
B) $\frac { 13 } { 3 }$
C) $\frac { 15 } { 4 }$
D) $\frac { 19 } { 6 }$
E) $\frac { 23 } { 6 }$