Math 151 Section 6.2
For
Area
f ( x )  0 on the interval [ a , b ] we approximate the area between the x-axis and the graph of
f (x ) . We fit thin rectangles using function values for heights and non-overlapping subintervals for
the bases. To describe this more clearly, we need some definitions.
A partition of [ a , b ] is a set of points
P
is
a  x0  x1  x2  ...  xn  b . The norm of the partition
= the length of the longest subinterval=
max ( xi  xi 1 )
i 1... n
The partition is regular if the lengths of the subintervals, [ xi 1 , xi ] , are all equal and hence each
subinterval length is
ba
.
n
In each subinterval [ xi 1 , xi ] choose any x-value, ci . Use
f ( ci ) for the height of the ith rectangle.
The ci ’s are called evaluation points.
The area of the ith rectangle is
f (ci )( xi  xi 1 ) . When we add these rectangle areas we have
n
 f (ci )( xi  xi 1 ) is an approximation to the area under the graph of f (x ) .
i 1
If
f (x ) is continuous on [ a , b ] , these sums approach the true area as P approaches 0.
Example 1 : Approximate the area under the graph of
regular partition and 4 subintervals. a) Use c1
f ( x)  x 2
over the interval [1, 3] using a
5
 1, c2  , c3  2, c4  3 as evaluation points. b)
3
Use the left endpoints for evaluation points. Sketch the graph and the rectangles.
c) Use the right endpoints for evaluations points. Sketch the graph and the rectangles.
A better approximation to either sum in parts b and c is their average.
Example where
0

 1
f ( x)  

 x
f (x ) is not continuous and the sums do not have a limit.
x0
x0
on [ 0,1]
Use n equal subintervals and use c1
evaluation points for the other rectangles.

1
n4
. Use any appropriate