To estimate the distance traveled by the vehicle over the 42-second interval, we'll use Riemann sums for approximating the area under the speed-time curve. This area represents the total distance traveled. We have three types of Riemann sums to calculate:

1. Left Riemann Sum (L6) – Using left endpoints.
2. Right Riemann Sum (R6) – Using right endpoints.
3. Midpoint Riemann Sum (M3) – Using midpoints with fewer intervals.

Each time interval, $\Delta t$, is 7 seconds.

Step 1: Left Riemann Sum (L6)

Using 6 intervals, we evaluate each interval's height based on the left endpoint.

$L6 = \Delta t \cdot (v(0) + v(7) + v(14) + v(21) + v(28) + v(35))$

Substitute the values: $L6 = 7 \cdot (0 + 8 + 27 + 48 + 61 + 59) = 7 \cdot 203 = 1421 \text{ ft}$

Step 2: Right Riemann Sum (R6)

Using 6 intervals, we evaluate each interval's height based on the right endpoint.

$R6 = \Delta t \cdot (v(7) + v(14) + v(21) + v(28) + v(35) + v(42))$

Substitute the values: $R6 = 7 \cdot (8 + 27 + 48 + 61 + 59 + 43) = 7 \cdot 246 = 1722 \text{ ft}$

Step 3: Midpoint Riemann Sum (M3)

Using 3 intervals, we evaluate each interval's height based on the midpoint of each interval.

The midpoints of each interval are:

1. From $t = 0$ to $t = 14$: midpoint is $t = 7$, so $v(7) = 8$
2. From $t = 14$ to $t = 28$: midpoint is $t = 21$, so $v(21) = 48$
3. From $t = 28$ to $t = 42$: midpoint is $t = 35$, so $v(35) = 59$

$M3 = 2 \cdot \Delta t \cdot (v(7) + v(21) + v(35))$

Substitute the values: $M3 = 14 \cdot (8 + 48 + 59) = 14 \cdot 115 = 1610 \text{ ft}$

Summary of Results:

• Left Riemann Sum (L6): $1421 \, \text{ft}$
• Right Riemann Sum (R6): $1722 \, \text{ft}$
• Midpoint Riemann Sum (M3): $1610 \, \text{ft}$

Would you like more details on any of these steps?

Related Questions

1. How is the trapezoidal rule different from the Riemann sums for estimating distances?
2. Why do we expect the Midpoint Riemann Sum to be more accurate than L6 and R6?
3. How would the result change if we used a different interval width?
4. Can Riemann sums be used to estimate the area under curves for non-uniform intervals?
5. What practical applications use Riemann sums beyond distance estimation?

Tip

When estimating areas under curves, Midpoint Riemann sums generally offer better accuracy because they balance the overestimation and underestimation across each interval.