1. **Problem Statement:** We have speeds of a bus measured every 10 minutes for an hour, and want to estimate total distance travelled using Riemann sums. 2. **Given:** Time intervals $t = 0, 10, 20, 30, 40, 50, 60$ minutes and speeds $v = 26, 40, 55, 10, 60, 32, 45$ km/h. 3. **Convert time intervals to hours:** Each subinterval is 10 minutes = $\frac{10}{60} = \frac{1}{6}$ hours. 4. **Define intervals and speeds:** - Intervals: $[0,10], [10,20], [20,30], [30,40], [40,50], [50,60]$ minutes - Speeds correspond to $v_0=26$, $v_1=40$, $v_2=55$, $v_3=10$, $v_4=60$, $v_5=32$, $v_6=45$ km/h. 5. **a. Left Riemann Sum, $L(6)$:** Uses left endpoints of each interval: $$ L(6) = \Delta t (v_0 + v_1 + v_2 + v_3 + v_4 + v_5) = \frac{1}{6}(26 + 40 + 55 + 10 + 60 + 32) $$ Calculate sum inside parentheses: $26 + 40 = 66$, $66 + 55 = 121$, $121 + 10 = 131$, $131 + 60 = 191$, $191 + 32 = 223$ Thus, $$ L(6) = \frac{1}{6} \times 223 = 37.1667\text{ km} $$ 6. **b. Right Riemann Sum, $R(6)$:** Uses right endpoints: $$ R(6) = \Delta t (v_1 + v_2 + v_3 + v_4 + v_5 + v_6) = \frac{1}{6}(40 + 55 + 10 + 60 + 32 + 45) $$ Calculate sum: $40 + 55 = 95$, $95 + 10 = 105$, $105 + 60 = 165$, $165 + 32 = 197$, $197 + 45 = 242$ Thus, $$ R(6) = \frac{1}{6} \times 242 = 40.3333\text{ km} $$ 7. **c. Midpoint Rule, $M(6)$:** Uses midpoints speed values. Midpoints occur at 5, 15, 25, 35, 45, 55 min. Estimate midpoint speeds by averaging speed at endpoints of each interval: - Interval 0-10: $(v_0 + v_1)/2 = (26 + 40)/2 = 33$ - Interval 10-20: $(v_1 + v_2)/2 = (40 + 55)/2 = 47.5$ - Interval 20-30: $(v_2 + v_3)/2 = (55 + 10)/2 = 32.5$ - Interval 30-40: $(v_3 + v_4)/2 = (10 + 60)/2 = 35$ - Interval 40-50: $(v_4 + v_5)/2 = (60 + 32)/2 = 46$ - Interval 50-60: $(v_5 + v_6)/2 = (32 + 45)/2 = 38.5$ Sum of midpoint speeds: $33 + 47.5 + 32.5 + 35 + 46 + 38.5 = 232.5$ Thus, $$ M(6) = \frac{1}{6} \times 232.5 = 38.75\text{ km} $$ **Final answers:** - Left Riemann Sum $L(6) = 37.17$ km - Right Riemann Sum $R(6) = 40.33$ km - Midpoint Rule $M(6) = 38.75$ km These estimates approximate the total distance travelled by the bus in 1 hour.