Hi. Need some guidance to make the trapezoid rule more effective.
Let be a function. I want to find an approximation to the integral
.
I know that the trapezoid rule is given by
where .
(Notice that I want trapezoids/parts)
can be rewritten as
.
Ok, now over to the problem. To make the method more effective I use the approximation to integral from in in this way
.
The part that I dont understand in this new formula is why you just can divide ? The other part I do understand, because in you have already calculated where is even and , so in you only need to calculate with odd indexes .
I hope the problem is clear!? And I hope someone can help me explain and understand this formula. Maby it's a good idea to prove that it works? I dont know how, but maby a hint could help?