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§ 6.2
Areas and Riemann Sums
Math 121 Lecture 30
!
Area Under a Graph
!
Riemann Sums to Approximate Areas (Midpoints)
!
Riemann Sums to Approximate Areas (Left Endpoints)
!
Applications of Approximating Areas

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*Sign up* Definition
Example
Area Under the Graph
of
f
(
x
)
from a to b
:
An
example of this is
shown to the right
In this section we will learn to
estimate
the area under the graph of
f
(
x
) from
x
=
a
to
x
=
b
by dividing up the interval into
partitions
(or subintervals),
each one having width
where
n
= the number of partitions that
will be constructed.
In the example below,
n
= 4.
A
Riemann Sum
is the sum of the areas of the rectangles generated above.

Use a Riemann sum to approximate the area under the graph
f
(
x
) on the given
interval using midpoints of the subintervals
The partition of -2
!
x
!
2 with
n
= 4 is shown below.
The length of each
subinterval is
-2
2
x
1
x
2
x
3
x
4
Observe the first midpoint is
units from the left endpoint, and the
midpoints themselves are
units apart.
The first midpoint is

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