Simpson’s 3/8 rule

Simpson’s $\frac{3}{8}$ rule is a method for approximating adefinite integral by evaluating the integrand at finitely manypoints. The formal rule is given by

$${\int }_{x_0}^{x_3}f(x)𝑑x\approx \frac{3h}{8}\left[f(x_0)+3f(x_1)+3f(x_2)+f(x_3)\right]$$

where $x_1=x_0+h$, $x_2=x_0+2h$, $x_3=x_0+3h$.

Simpson’s $\frac{3}{8}$ rule is the third Newton-Cotes quadratureformula. It has degree of precision 3. This means it is exact forpolynomials of degree less than or equal to three. Simpson’s$\frac{3}{8}$ rule is an improvement to the traditional Simpson’srule. The extra function evaluation gives a slightly more accurateapproximation . We can see this with an example.

Using the fundamental theorem of the calculus, one shows

$${\int }_0^{\pi }\sin (x)𝑑x=2.$$

In this case Simpson’s rule gives,

$${\int }_0^{\pi }\sin (x)𝑑x\approx \frac{\pi }{6}\left[\sin (0)+4\sin \left(\frac{\pi }{2}\right)+\sin (\pi )\right]=\mathrm{\hspace{0.17em}2.094}$$

However, Simpson’s $\frac{3}{8}$ rule does slightly better.

$${\int }_0^{\pi }\sin (x)𝑑x\approx \left(\frac{3}{8}\right)\frac{\pi }{3}\left[\sin (0)+3\sin \left(\frac{\pi }{3}\right)+3\sin \left(\frac{2\pi }{3}\right)+\sin (\pi )\right]=\mathrm{\hspace{0.17em}2.040}$$