Math · Calculus I · Worked example
Estimate distance from a table of speeds
A car’s speed is read every 2 seconds: 0, 6, 10, 13 and 15 m/s at t = 0, 2, 4, 6 and 8 s. Estimate the distance it travels in those 8 seconds.
Read the width and the heights
Distance is the integral of speed, so a Riemann sum estimates it. The readings are Δt = 2 s apart, giving four subintervals.
| t (s) | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| v (m/s) | 0 | 6 | 10 | 13 | 15 |
Left sum
Use the first four readings, one at the start of each interval.
Right sum
Use the last four readings, one at the end of each interval.
Average them
The speed rises at every reading. If it rose steadily between readings too, the true distance lies between 58 m and 88 m. The trapezoid estimate is their average.
Result
Between 58 m and 88 m; the trapezoid estimate is 73 m.
Your turn
Water flows into a tank at 12, 10, 7, 5 and 4 L/min at t = 0, 5, 10, 15 and 20 min. Estimate the water added in 20 minutes with a right sum. Is it an over- or underestimate?
Show the answer and explanation
130 L, an underestimate if the rate fell steadily.
Δt = 5 min and the right sum uses 10, 7, 5 and 4: 5(10 + 7 + 5 + 4) = 130 L. The rate falls, so each right-end height is the smallest in its interval and the sum is too low. The left sum, 170 L, is too high.
Keep exploring
The right sum minus the left sum is Δt times the total change in speed: 2 × 15 = 30 m. Readings every second would halve that gap to 15 m.
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