Which type of Riemann Sum is the most accurate?

Question

When estimating the area under a curve, is it more accurate to average out the values of the upper and lower sums or use the midpoints of the intervals instead?

I would assume it varies but I just wanted to know what the best way is to estimate the area under a curve when no instructions on how to do so have been given.

All the methods you have listed are only estimates. None of them are exact. The 'best' way of finding the area is integration. — Landuros, Jan 14 '18 at 03:40

Mohammad Riazi-Kermani · Accepted Answer · 2018-01-14T04:22:27.220

10

Your question is about the error analysis of Numerical Methods of approximating a definite integral.

Both midpoint rule and trapezoidal rule have a global error which is $O(h^2)$ where $h=(b-a)/n$ is the step size.

That means if you double your $n$ the error divides by $4$.

Among the midpoint and the trapezoidal rules, the midpoint is the better choice because its error estimate is half of the trapezoidal rule.

The error estimate for trapezoidal rule is $$ -\frac {(b-a)h^2}{12}f''(\eta)$$While the error estimate for the midpoint rule is $$ \frac {(b-a)h^2}{24}f''(\eta)$$where $\eta \in [a,b]$ depends on the function.

edited Jan 14 '18 at 04:22

answered Jan 14 '18 at 04:08

Mohammad Riazi-Kermani

69,891

4

Note that the two $\eta$'s are different in general. While midpoint is "usually" better, there are cases where the trapezoidal rule turns out to be better. – Robert Israel Jan 14 '18 at 06:03
The midpoint error can be larger than the trapezoidal error when the integrand exhibits a high degree of curvature in a small subinterval. See here. – RRL Jan 14 '18 at 07:07
There are other quadrature methods that converge even faster. – Wouter Jan 14 '18 at 09:42
Yes, Simpson Method, comes next. Gaussian quadrature is a jem and many Gaussian type quadratures exist. – Mohammad Riazi-Kermani Jan 14 '18 at 10:33

Which type of Riemann Sum is the most accurate?

1 Answers1