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.
Best Answer
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.