230
4.2. RIEMANN SUMS
Finally, we recall from the introduction to this present section that in the context
where the function f represents the velocity of a moving object, the total sum of the areas
bounded by the curve tells us the total distance traveled over the relevant time interval,
while the total net signed area bounded by the curve computes the object’s change in
position on the interval.
Activity 4.6.
Suppose that an object moving along a straight line path has its velocity v (in feet per
second) at time t (in seconds) given by
v(t) =
1
2
t
2 − 3t +
7
2
.
(a) Compute M 5 , the middle Riemann sum, for v on the time interval [1, 5]. Be
sure to clearly identify the value of △t as well as the locations of t 0 , t 1 , · · · , t 5 .
In addition, provide a careful sketch of the function and the corresponding
rectangles that are being used in the sum.
(b) Building on your work in (a), estimate the total change in position of the object
on the interval [1, 5].
(c) Building on your work in (a) and (b), estimate the total distance traveled by the
object on [1, 5].
(d) Use appropriate computing technology 5 to compute M 10 and M 20 . What exact
value do you think the middle sum eventually approaches as n increases without
bound? What does that number represent in the physical context of the overall
problem?
⊳
Summary
In this section, we encountered the following important ideas:
• A Riemann sum is simply a sum of products of the form f (x ∗
i )△x that estimates the
area between a positive function and the horizontal axis over a given interval. If
the function is sometimes negative on the interval, the Riemann sum estimates the
difference between the areas that lie above the horizontal axis and those that lie below
the axis.
• The three most common types of Riemann sums are left, right, and middle sums, plus
we can also work with a more general, random Riemann sum. The only difference
5 For instance, consider the applet at http://gvsu.edu/s/a9 and change the function and adjust the
locations of the blue points that represent the interval endpoints a and b.
4.2. RIEMANN SUMS
Finally, we recall from the introduction to this present section that in the context
where the function f represents the velocity of a moving object, the total sum of the areas
bounded by the curve tells us the total distance traveled over the relevant time interval,
while the total net signed area bounded by the curve computes the object’s change in
position on the interval.
Activity 4.6.
Suppose that an object moving along a straight line path has its velocity v (in feet per
second) at time t (in seconds) given by
v(t) =
1
2
t
2 − 3t +
7
2
.
(a) Compute M 5 , the middle Riemann sum, for v on the time interval [1, 5]. Be
sure to clearly identify the value of △t as well as the locations of t 0 , t 1 , · · · , t 5 .
In addition, provide a careful sketch of the function and the corresponding
rectangles that are being used in the sum.
(b) Building on your work in (a), estimate the total change in position of the object
on the interval [1, 5].
(c) Building on your work in (a) and (b), estimate the total distance traveled by the
object on [1, 5].
(d) Use appropriate computing technology 5 to compute M 10 and M 20 . What exact
value do you think the middle sum eventually approaches as n increases without
bound? What does that number represent in the physical context of the overall
problem?
⊳
Summary
In this section, we encountered the following important ideas:
• A Riemann sum is simply a sum of products of the form f (x ∗
i )△x that estimates the
area between a positive function and the horizontal axis over a given interval. If
the function is sometimes negative on the interval, the Riemann sum estimates the
difference between the areas that lie above the horizontal axis and those that lie below
the axis.
• The three most common types of Riemann sums are left, right, and middle sums, plus
we can also work with a more general, random Riemann sum. The only difference
5 For instance, consider the applet at http://gvsu.edu/s/a9 and change the function and adjust the
locations of the blue points that represent the interval endpoints a and b.