Calculus- Early Transcendentals, 2021a

226 Integration
Figure 6.9: General Riemann sum to approximate the area under f (x)=4x − x 2 .
Figure 6.9 shows the approximating rectangles of a Riemann sum. While the rectangles in this example
do not approximate well the shaded area, they demonstrate that the subinterval widths may vary and the
heights of the rectangles can be determined without following a particular rule.
Riemann sums are typically calculated using one of the three rules we have introduced. The uniformity
of construction makes computations easier. Before working another example, let’s summarize some of
what we have learned in a convenient way.
Riemann Sums
Consider a function f (x) defined on an interval [a,b]. The area under this curve is approximated by
∑ n i=1 f (c i)Δx i .
1. When the n subintervals have equal length, Δx i = Δx = b − a
n .
2. The i th term of the partition is x i = a +(i − 1)Δx. (Thismakesx n+1 = b.)
n
3. The Left Hand Rule summation is: ∑ f (x i )Δx.
i=1
4. The Right Hand Rule summation is:
5. The Midpoint Rule summation is:
n
∑ f (x i+1 )Δx.
i=1
(
xi + x x+1
n
∑ f
i=1
2
)
Δx.
Let’s do another example.
Example 6.9: Approximating Area Using Sums
Approximate the area under f (x)=(5x +2) on the interval [−2,3] using the Midpoint Rule and ten
equally spaced intervals.
Solution. Following the above discussion, we have
Δx = 3 − (−2) = 1/2
10