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4.2. RIEMANN SUMS
estimate of the area under the curve y = f (x) over the interval [a, b]; momentarily, we will
discuss the meaning of Riemann sums in the setting when f is sometimes negative. We also
recall that in the context of a nonnegative velocity function y = v(t), the corresponding
Riemann sums are approximating the distance traveled on [a, b] by the moving object with
velocity function v.
There is a more general way to think of Riemann sums, and that is to not restrict the
choice of where the function is evaluated to determine the respective rectangle heights.
That is, rather than saying we’ll always choose left endpoints, or always choose midpoints,
we simply say that a point x ∗
i+1 will be selected at random in the interval [x i , x i+1 ] (so that
x i ≤ x ∗
i+1 ≤ x i+1 ), which makes the Riemann sum given by
f (x
∗
1 ) · △x + f (x
∗
2 ) · △x + · · · + f (x
∗
i+1 ) · △x + · · · + f (x
∗
n ) · △x =
n
i=1
f (x
∗
i )△x.
At http://gvsu.edu/s/a9, the applet noted earlier and referenced in Figure 4.15, by
unchecking the “relative” box at the top left, and instead checking “random,” we can easily
explore the effect of using random point locations in subintervals on a given Riemann sum.
In computational practice, we most often use L n , R n , or M n , while the random Riemann
sum is useful in theoretical discussions. In the following activity, we investigate several
different Riemann sums for a particular velocity function.
Activity 4.5.
Suppose that an object moving along a straight line path has its velocity in feet per
second at time t in seconds given by v(t) =
2
9 (t − 3) 2 + 2.
(a) Carefully sketch the region whose exact area will tell you the value of the
distance the object traveled on the time interval 2 ≤ t ≤ 5.
(b) Estimate the distance traveled on [2, 5] by computing L 4 , R 4 , and M 4 .
(c) Does averaging L 4 and R 4 result in the same value as M 4 ? If not, what do you
think the average of L 4 and R 4 measures?
(d) For this question, think about an arbitrary function f , rather than the particular
function v given above. If f is positive and increasing on [a, b], will L n overestimate or under-estimate the exact area under f on [a, b]? Will R n over- or
under-estimate the exact area under f on [a, b]? Explain.
⊳
When the function is sometimes negative
For a Riemann sum such as
L n =
n−1
i=0
f (x i )△x,
4.2. RIEMANN SUMS
estimate of the area under the curve y = f (x) over the interval [a, b]; momentarily, we will
discuss the meaning of Riemann sums in the setting when f is sometimes negative. We also
recall that in the context of a nonnegative velocity function y = v(t), the corresponding
Riemann sums are approximating the distance traveled on [a, b] by the moving object with
velocity function v.
There is a more general way to think of Riemann sums, and that is to not restrict the
choice of where the function is evaluated to determine the respective rectangle heights.
That is, rather than saying we’ll always choose left endpoints, or always choose midpoints,
we simply say that a point x ∗
i+1 will be selected at random in the interval [x i , x i+1 ] (so that
x i ≤ x ∗
i+1 ≤ x i+1 ), which makes the Riemann sum given by
f (x
∗
1 ) · △x + f (x
∗
2 ) · △x + · · · + f (x
∗
i+1 ) · △x + · · · + f (x
∗
n ) · △x =
n
i=1
f (x
∗
i )△x.
At http://gvsu.edu/s/a9, the applet noted earlier and referenced in Figure 4.15, by
unchecking the “relative” box at the top left, and instead checking “random,” we can easily
explore the effect of using random point locations in subintervals on a given Riemann sum.
In computational practice, we most often use L n , R n , or M n , while the random Riemann
sum is useful in theoretical discussions. In the following activity, we investigate several
different Riemann sums for a particular velocity function.
Activity 4.5.
Suppose that an object moving along a straight line path has its velocity in feet per
second at time t in seconds given by v(t) =
2
9 (t − 3) 2 + 2.
(a) Carefully sketch the region whose exact area will tell you the value of the
distance the object traveled on the time interval 2 ≤ t ≤ 5.
(b) Estimate the distance traveled on [2, 5] by computing L 4 , R 4 , and M 4 .
(c) Does averaging L 4 and R 4 result in the same value as M 4 ? If not, what do you
think the average of L 4 and R 4 measures?
(d) For this question, think about an arbitrary function f , rather than the particular
function v given above. If f is positive and increasing on [a, b], will L n overestimate or under-estimate the exact area under f on [a, b]? Will R n over- or
under-estimate the exact area under f on [a, b]? Explain.
⊳
When the function is sometimes negative
For a Riemann sum such as
L n =
n−1
i=0
f (x i )△x,