Speedometer readings for a motorcycle at 12-seconds intervals are given in the table.

Question

Speedometer readings for a motorcycle at 12-seconds intervals are given in the table.

t (sec) 0 12 24 36 48 60

v (ft/sec) 22 26 29 23 23 30

(a) Estimate the distance, in feet, traveled by the motorcycle during the time interval from 0 to 60 seconds using the velocities at the beginning of the time intervals.



(b) Give another estimate using the velocities at the end of the time periods.

Answer 1

So this question is effectively asking for a comparison of the right and left Riemann sums from the data in the table. You just need to remember that the area under the graph of the velocity represents the distance traveled.

But even without needing to know that, all you need is distance = velocity * time.

So total distance = velocity₀ * Δt + velocity₀ * Δt + velocity₁₂ * Δt + velocity₂₄ * Δt + velocity₃₆ * Δt + velocity₄₈ * Δt

_{Where Δt = time from that velocity to the next data point}

This total distance uses the velocity starting from the right. Using the velocities at the end of the time period would look exactly the same, just changing velocity₀ to velocity₆₀

Then it's just a matter of multiplication and addition to find the estimated total distance traveled, yielding 1476 feet and 1572 for beginning of the time interval and end, respectively