Riemann Sum
Let a closed
interval ![[a,b]](Riemann%20Sum%20--%20from%20Wolfram%20MathWorld_files/Inline1.gif)
be partitioned by points 
, where the lengths of the resulting intervals between the
points are denoted 
, 
, ..., 
. Let 
be an arbitrary point in the 
th subinterval. Then the quantity
 |
is called a Riemann sum for a given function 
and partition, and the value 
is called the mesh
size of the partition.
If the limit of
the Riemann sums exists as 
, this limit is known as the Riemann integral of 
over the interval ![[a,b]](Riemann%20Sum%20--%20from%20Wolfram%20MathWorld_files/Inline12.gif)
. The shaded areas in the above plots show the lower
and upper sums
for a constant mesh
size.
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