Context: high school math
Question: integrating a semicircle $y= \sqrt{1-x^2}$ from $0$ to $1$ using five function values with Simpson’s rule gives me $0.7709$. The exact value is $0.785$.
I understand for the trapezoidal rule we can predict if it under/overestimates the area based on the curve’s concavity.
So I wonder if there's a simple way to predict whether Simpson’s rule over/underestimates a curve like this? (I understand Simpson’s rule holds equality for linear and cubic equations)
Edited to add: I have used the formula for error in the comments below and I’m struggling with finding the maximum absolute value of the fourth derivative since there is a vertical asymptomate at x=1 there, please see below graphs of first, second, third and fourth derivatives of the curve
