a.) Use six rectangles to find estimates in the following sample points for the area under the given graph of $f$ from $x = 0$ to $x = 12$.

By Dividing the interval into six sub interval..

$\displaystyle \Delta x = \frac{12 - 0}{6} = 2$

(i) Left endpoints $L_6$

$ \begin{equation} \begin{aligned} L_6 =& \sum^6_1 f(x_1) \Delta x \\ \\ L_6 =& 2 [f(0) + f(2) + f(4) + f(6) + f(8) + f(10)] \\ \\ L_6 =& 2 [9 + 8.8 + 8.2 + 7.3 + 5.9 + 4.1] \\ \\ L_6 =& 86.6 \end{aligned} \end{equation} $

(ii) Right endpoints $R_6$

$ \begin{equation} \begin{aligned} R_6 =& \sum^6_{i = 1} f(xi) \Delta x \\ \\ R_6 =& 2 [f(2) + f(4) + f(6) + f(8) + f(10) + f(12)] \\ \\ R_6 =& 2 [8.8 + 8.2 + 7.3 + 5.9 + 4.1 + 1] \\ \\ R_6 =& 70.6 \end{aligned} \end{equation} $

(iii) Midpoints $M_6$

$ \begin{equation} \begin{aligned} M_6 =& \sum \limits_{i = 1}^6 f(xi) \Delta x \\ \\ M_6 =& 2 [f(1) + f(3) + f(5) + f(7) + f(9) + f(11)] \\ \\ M_6 =& 2 [8.9 + 8.5 + 7.8 + 6.6 + 5.0 + 2.8] \\ \\ M_6 =& 79.2 \end{aligned} \end{equation} $

b.) Is $L_6$ an underestimate or overestimate of the true area?

$L_6$ is an overestimate of the true area since the function is decreasing and the bars we used are over the graph.

c.) Is $R_6$ an underestimate or overestimate of the true area?

$R_6$ is an underestimate of the true area. The bars we used are always under the graph.

d.) Which of the numbers $L_6, R_6$ or $M_6$ gives the best estimate? Explain.

$M_6$ gives the best estimate because the area of each rectangular bar appears to be closer to the true area compare to $L_6$ and $R_6$