8. Speedometer readings for a motorcycle at 12 -second...

8. Speedometer readings for a motorcycle at 12 -second intervals are given in the table. (a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals. (b) Give another estimate using the velocities at the end of the time periods. (c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain. $$\begin{array}{|c|c|c|c|c|c|c|}\hline t(s) & {0} & {12} & {24} & {36} & {48} & {60} \\ \hline v(f t / s) & {30} & {28} & {25} & {22} & {24} & {27} \\ \hline\end{array}$$

8. Speedometer readings for a motorcycle at 12 -second
intervals are given in the table.
(a) Estimate the distance traveled by the motorcycle during
this time period using the velocities at the beginning of
the time intervals.
(b) Give another estimate using the velocities at the end of
the time periods.
(c) Are your estimates in parts (a) and (b) upper and lower
estimates? Explain.
$$\begin{array}{|c|c|c|c|c|c|c|}\hline t(s) & {0} & {12} & {24} & {36} & {48} & {60} \\ \hline v(f t / s) & {30} & {28} & {25} & {22} & {24} & {27} \\ \hline\end{array}$$

Key Concepts

Right-Hand Riemann Sum

The right-hand Riemann sum is another approach to approximating the area under a curve where the height of each rectangle is taken from the right endpoint of each subinterval. Like the left-hand sum, it provides an approximation of the definite integral, but the error direction (under or overestimate) will depend on whether the function is increasing or decreasing over the subintervals.

Upper and Lower Estimates

Upper and lower estimates refer to the tendencies of certain numerical integration methods to either overestimate or underestimate the actual value of an integral. When using left-hand or right-hand Riemann sums, the curvature and monotonicity of the function within each interval determine whether the sum acts as an upper or lower bound for the true area under the curve.

Riemann Sum Method

The Riemann sum is a fundamental concept in numerical integration where a function is approximated by summing the areas of rectangles whose heights are determined by the function's values at specific sample points. This method serves as an approximation for calculating definite integrals, especially when an analytical solution is difficult or when data is given in discrete intervals.

Left-Hand Riemann Sum

The left-hand Riemann sum is a specific type of Riemann sum where the height of each rectangle is determined by the value of the function at the left endpoint of each subinterval. This method approximates the area under the curve by using values that occur at the beginning of each interval, and depending on the function's behavior, it tends to produce an underestimate or overestimate.

Distance as the Area Under a Speed-Time Graph

In the context of motion, the distance traveled by an object can be approximated by calculating the area under its speed versus time graph. By applying numerical integration methods such as Riemann sums, one sums the product of the speed values and the duration of the time intervals to estimate the total distance covered.

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