It is appropriate to spend some time considering what sorts of
errors are present in an algorithm such as the two integration
methods. Error analysis is based on a Taylor expansion, with remainder.
For any point x, consider a nearby (fixed) point, a.
We can expand
, where
is some point between x and a.
For specificity, consider a subinterval with left endpoint a and
right endpoint b. To approxiamate the error in the subinterval,
(I am approximating 
.)
So in each subinterval, the trapezodial rule makes an error
(this is the local truncation error).
There are N such subintervals
so the net error in the integration is approximately
(global truncation error).
That is, the
trapezodial rule is second-order accurate.
Note: Using just a rectangle based on right or left endpoints,
the Taylor exapansion reads 
. The
error then goes like 
, i.e. first-order accuracy.
Now look at your results from your trapezodial integration
assignment. For 
, you can do the integration
exactly. Look at the error you made using 10, 20, and 40 subintervals;
what is the relation of these errors?
Look at the handout on Approximate Convergence Order, and the
accompanying code. Are you getting the correct order of
convergence?