42. Calculate the approximate area of the shaded region in the figure by the trapezoidal rule, using divisions at $x = \frac { 4 } { 3 }$ and $x = \frac { 5 } { 3 }$.
(A) $\frac { 50 } { 27 }$
(B) $\frac { 251 } { 108 }$
(C) $\frac { 7 } { 3 }$
(D) $\frac { 127 } { 54 }$
(E) $\frac { 77 } { 27 }$

18. If three equal subdivisions of $[ - 4,2 ]$ are used, what is the trapezoidal approximation of $\int _ { - 4 } ^ { 2 } \frac { e ^ { - x } } { 2 } d x ?$
(A) $e ^ { 2 } + e ^ { 0 } + e ^ { - 2 }$
(B) $e ^ { 4 } + e ^ { 2 } + e ^ { 0 }$
(C) $e ^ { 4 } + 2 e ^ { 2 } + 2 e ^ { 0 } + e ^ { - 2 }$
(D) $\frac { 1 } { 2 } \left( e ^ { 4 } + e ^ { 2 } + e ^ { 0 } + e ^ { - 2 } \right)$
(E) $\frac { 1 } { 2 } \left( e ^ { 4 } + 2 e ^ { 2 } + 2 e ^ { 0 } + e ^ { - 2 } \right)$

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.)

124- What is the value of $\displaystyle\int_1^4 \sqrt{\left(\frac{1}{2}x^2 - \frac{1}{x}\right)^2 + 1}\,dx$?
$4\ (1$ $5\ (2$ $6\ (3$ $7\ (4$

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

Use the trapezium rule with 3 strips to estimate
$$\int _ { \frac { 1 } { 2 } } ^ { 2 } 2 \log _ { 10 } x \mathrm {~d} x$$
A $\log _ { 10 } \frac { \sqrt { 6 } } { 2 }$
B $\log _ { 10 } \frac { 3 } { 2 }$
C $\log _ { 10 } \frac { 9 } { 4 }$
D $\log _ { 10 } 3$
E $\quad \log _ { 10 } \frac { 81 } { 16 }$
F $\log _ { 10 } \frac { \sqrt { 23 } } { 2 }$

The trapezium rule with 4 strips is used to estimate the integral:
$$\int _ { - 2 } ^ { 2 } \sqrt { 4 - x ^ { 2 } } d x$$
What is the positive difference between the estimate and the exact value of the integral?

$$\begin{aligned} & f : [ 1,3 ] \rightarrow [ 2,10 ] \\ & f ( x ) = 1 + x ^ { 2 } \end{aligned}$$
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) $\frac { 23 } { 6 }$