Chapter 4: Integration
Section 4.1: Area by Riemann Sums
If fx≥0, the area bounded by the x-axis, and the graph of f on the interval a≤x≤b, is defined to be the limiting value of the Riemann sum
where x0=a, xn=b, and xk,k=1,…,n−1, form a partition of the interval a,b, with Δxk=xk−xk−1, and xk∗∈xk−1,xk.
Figure 4.1.1 Area "under" the graph of fx
Table 4.1.1 lists six common types of Riemann sums.
xk∗ are chosen so that each fxk∗ is a maximum on xk−1,xk
sum is a maximum for that partition
xk∗ are chosen so that each fxk∗ is a minimum on xk−1,xk
sum is a minimum for that partition
xk∗ are chosen as the midpoints of each subinterval xk−1,xk
xk∗=xk−1 (evaluate f at the left end of each subinterval)
xk∗=xk (evaluate f at the right end of each subinterval)
xk∗ are selected at random in each subinterval xk−1,xk
Table 4.1.1 Six types of Riemann sums
The limiting process requires n→∞ and maxkΔxk→0. (The number of subintervals in a,b must become infinite, and the width of the widest subinterval must shrink to zero.)
If f is continuous on a,b, the limit exists and is independent of the partition.
The simplest Riemann sums are those in which the xk are equispaced so that Δxk=b−an=h, k=1,…,n.
Use a left Riemann sum to obtain the area bounded by the graph of fx=6+x−x2 and the x-axis.
Use a Riemann sum to obtain the area bounded by the graph of fx=x3−2⁢x2−5 x+6 and the x-axis.
Use Maple to obtain −125n3∑i=1ni2−i⁢n as the right Riemann sum for fx=6+x−x2, −2≤x≤3.
Use Maple to evaluate this Riemann sum to 1256 n2−1n2.
Obtain closed-form expressions for ∑k=1na, ∑k=1nk, and ∑k=1nk2; then use these expressions to show how the right Riemann sum becomes 1256 n2−1n2.
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