When computing exact integrals, we have an infinite number of intervals and h is infinitely small (very close to 0), so if h is the length of the intervals, why wouldn't the error be O(h) or O(h*#number of intervals)?

ddkim

O(1/n^2) is a big-O expression I've never really seen before - how does it match up to other big-O expressions in terms of "speed"?

large_monkey

I wonder if there are pathological examples that are continuous, but perhaps not even differentiable, where the trapezoidal rule can be made to be arbitrarily bad? What would these look like?

Why 1/n^2?

When computing exact integrals, we have an infinite number of intervals and h is infinitely small (very close to 0), so if h is the length of the intervals, why wouldn't the error be O(h) or O(h*#number of intervals)?

O(1/n^2) is a big-O expression I've never really seen before - how does it match up to other big-O expressions in terms of "speed"?

I wonder if there are pathological examples that are continuous, but perhaps not even differentiable, where the trapezoidal rule can be made to be arbitrarily bad? What would these look like?