The approximations
and 
are not very accurate on large intervals (error varying as
and
respectively but more accurate approximations can be obtained by breaking the interval
into
subintervals of equal width
Composite Trapezium Rule
Split the intervalintointervals of width
i.e. consider
equispaced
points
The subinterval
is
But since
and f'' is continuous, there exists
with
so the error is of the form
since
We can write the composite trapezium rule as
The error is of order
compared to
for each subinterval.