4.3 RIEMANN SUM AND DEFINITE INTEGRALS
A Partition with subintervals of n equal lengths
Find the area bounded by the graph of
f(x) = x
and the x-axis for x on [0, 1].
Sketch the graph:
List right end-points in form of i.
List the heights in form of i.
Write the widths in form of i.
Write the area using Sigma Notation.
Find the actual area by finding limit of the sum as n approach
DEFINITION OF THE RIEMANN SUM:
Let f be defined on a closed interval [a, b], and let f be defined on a closed interval [a, b], and let be a partition
[a, b] given: a = x
=b where x
is the length of the i
is any point in the i
subinterval, then the sum
Is called the Riemann Sum of f for the partition .
If f is defined on the closed interval [a, b] and the limit:
exists, then f is integrable on [a, b] and the limit is denoted:
The limit is called the definite integral of f from a to b.
The number a is the lower limit of the