Approximating definite integrals
To approximate the definite integral $\int_a^b f(x) dx$ we have different means, including
(right) Riemann sum: $I \approx f(b)\cdot (b-a)$
trapezoid method: $I \approx 1/2 \cdot (f(b) + f(a)) \cdot (b-a)$
Simpson's rule: $I \approx (b-a)/6 \cdot (f(a) + 4f(a/2 + b/2)) + f(b)$
Gauss 7-point quadrature $I \approx \sum w_k f(n_k)$ for selected weights and nodes.
The first 3 are applied below to a partition of $[a,b]$ into $n$ equal-sized intervals. The latter adaptively used in quadgk
$\int_0^\pi \sin(x) dx = 2$.
Using 7-point Gauss on $[0, \pi]$: 2.0000000000017906
Using quadgk n $[0, \pi]$: 2.000000021101344
Which method =
$n =$