Is it possible to determine which yields a better approximation for $\pi$?

Question

If I use the trapezoidal rule, using two equal partitions, to estimate $$\int_{0.5}^{1} \sqrt{1-x^2}dx$$ I can obtain an approximation of $\pi$, as the integral's exact value can be found without much difficulty.

Say I use the trapezoidal rule with 4 equal partitions to estimate $$\int_{0}^{1} \sqrt{1-x^2}dx$$

and thus obtain another approximation for $\pi$.

Without explicitly finding the both of them and seeing which is more accurate, is there some sort of conceptual way of determining which will yield the better approximation? I can see that the second method, using 4 equal partitions, contains the two partitions from the first method. However, I am not completely sure as to how this can help us.

Answer 1

Think of it this way, as you use an approximation there is an error , $\epsilon$ when you use more partition and expand the boundaries equally, you'll accumulate much more errors, so you should expect that the first one should be precise but anyway it depends on how the curve curves !, I'm new here, don't know how to comment/sorry !

Answer 2

Better than either way would be to estimate the integral from $0$ to $\frac12$, using two equal partitions as before. The reason is that the errors arise from omitting the segments of the circle cut off by the slanting sides at the top of the trapezia. The greater the slant (as with the sides on the right), the longer the side, and so the greater the omitted segment.

If you must use one or other of the ranges $0$ to $1$ or $\frac12$ to $1$, then the former would be preferable, because the larger error in the latter range would be diluted in its combination with the lower-error range $0$ to $\frac12$.