Calculus- Early Transcendentals, 2021a

6.1. Displacement and Area 225
We do so here, skipping from the original summand to the equivalent of Equation 6.1 to save space.
Note that Δx = 4/1000 = 0.004.
1000
∑
i=1
f (x i+1 )Δx =(4Δx 2 1000
)
∑
i=1
i − Δx 3 1000
∑ i 2
i=1
=(4Δx 2 1000 · 1001
) − Δx 3 1000(1001)(2001)
2
6
= 4 · 0.004 2 · 500500 − 0.004 3 · 333,833,500
= 10.666656
Using many, many rectangles, we have a likely good approximation of the area under f (x)=4x − x 2
of ≈ 10.666656.
♣
Before the above example, we stated the summations for the Left Hand, Right Hand and Midpoint
Rules. Each had the same basic structure, which was:
1. each rectangle has the same width, which we referred to as Δx, and
2. each rectangle’s height is determined by evaluating f (x) at a particular point in each subinterval.
For instance, the Left Hand Rule states that each rectangle’s height is determined by evaluating f (x)
at the left hand endpoint of the subinterval the rectangle lives on.
One could partition an interval [a,b] with subintervals that did not have the same width. We refer to
the length of the first subinterval as Δx 1 , the length of the second subinterval as Δx 2 , and so on, giving the
length of the i th subinterval as Δx i . Also, one could determine each rectangle’s height by evaluating f (x) at
any point in the i th subinterval. We refer to the point picked in the first subinterval as c 1 , the point picked
in the second subinterval as c 2 , and so on, with c i representing the point picked in the i th subinterval. Thus
the height of the i th subinterval would be f (c i ),andtheareaofthei th rectangle would be f (c i )Δx i .
Summations of rectangles with area f (c i )Δx i are named after mathematician Georg Friedrich Bernhard
Riemann, as given in the following definition.
Definition 6.8: Riemann Sum
Let f (x) be defined on the closed interval [a,b] and let Δx be a partition of [a,b], with
a = x 1 < x 2