where
and
the midpoint of the subinterval .
where
and
where
and
Error Bounds
where K is an upper-bound for on [a,b].
where K is an upper-bound for on [a,b].
where K is an upper-bound for on [a,b].
Upper-bounds might be determined algebraically, estimated graphically, or derived from max/min considerations. Note that K is different in the different formulas.
Recall the definitions of the Riemann sum: the sum
where , named after Bernhard Riemann (1826-1866), a student of Gauss.
In general
There are two major motivations for approximate integration:
has no ``closed-form'' solution (can't find an anti-derivative); and
The choices for are usually step-functions (Left, Right, and Midpoint Rectangle rules), or continuous but non-differentiable functions (Trapezoidal and Simpson's rule). Other (better!) rules use continuous and smooth functions.
The trapezoidal rule is simply the average of the left and right rectangle rules, a primitive improvement on them both. It is also equivalent to adding up the area under the trapezoids created by connecting left and right endpoints of the curve and then dropping the ends down to the x-axis.
The midpoint rule is created on the hope that we can avoid extremes at the left and right endpoints. It is another simple rectangle rule, but evidently generally better than the trapezoidal rule.
Simpson's rule is derived using ``best-fitting'' parabolas, rather than straight line segments. It is considered a very good method in general.
All the error bounds rely on having a bound on a higher derivative of the function. This may not be easy to obtain.