For questions regarding Simpson's rule for approximating the integral of a function using quadratic polynomials, and its applications.
Simpson's rule is a technique for estimating integrals numerically, and can be written as
$$\int_a^b f(x) dx \approx \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]$$
Simpson's rule can be derived by using a quadratic approximation for $f$, or it can be viewed as a weighted average of the trapezoid and midpoint rules. The error involved can be bounded above by
$$\frac{1}{90} \left(\frac{b - a}{2}\right)^5 \left|f^{(4)}(\xi)\right|$$
where $\xi$ is a number between $a$ and $b$.
Reference: Simpson's rule.
