Consider the function $f(x)=\sin(x^2/2)$ on the interval $[0,1],$ and let $I=\int_{0}^{\,1} {f(x)}\, dx.$

(a) By looking at a sketch of the graph of $f(x),$ determine for each estimate whether it overestimates, underestimates, or is equal to $I$.

? Underestimate Overestimate Exact  1. $T_2$

? Underestimate Overestimate Exact  2. $M_2$

? Underestimate Overestimate Exact  3. $R_2$

? Underestimate Overestimate Exact  4. $L_2$

(b) For any value of $n$, list the numbers $L_n,$ $R_n,$ $M_n,$ $T_n,$ and $I$ in increasing order. For each of the five values, order them from 1 to 5, with 1 indicating the smallest of the five values and 5 being the largest of the values.

? 1 2 3 4 5  1. $T_n$

? 1 2 3 4 5  2. $I$

? 1 2 3 4 5  3. $L_n$

? 1 2 3 4 5  4. $R_n$

? 1 2 3 4 5  5. $M_n$

(d) Compute $L_5,$ $R_5,$ $M_5,$ and $T_5$. Give each answer correct to five decimal places.

$L_5 =$

$R_5 =$

$M_5 =$

$T_5 =$

(e) From the graph, which do you think gives the best estimate of $I?$

A. $L_5$
B. $T_5$
C. $R_5$
D. $M_5$