Simpson 3/8

Description

Given a continuous function f:[a,b]→Rf: [a,b] \rightarrow \Reals, we want to calculate:

∫abf(x)\int_{a}^{b} f(x)

If FF is a primitive of ff:

∫abf(x)dx=F(b)−F(a)\int_a^b f(x)dx = F(b) - F(a)

Since there is not always a known primitive FF, it’s better to approximate the value of the integral. The Simpson’s 38\frac{3}{8} states that:

∫abf(x) dx≈38h[f(a)+3f(2a+b3)+3f(a+2b3)+f(b)]≈b−a8[f(a)+3f(2a+b3)+3f(a+2b3)+f(b)]\begin{aligned} \int_a^b f(x) \, dx & \approx \frac{3}{8} h \left[ f(a) + 3f\left(\frac{2a + b}{3}\right) + 3f\left(\frac{a + 2b}{3}\right) + f(b) \right] \\ & \approx \frac{b - a}{8} \left[ f(a) + 3f\left(\frac{2a + b}{3} \right) + 3f\left(\frac{a + 2b}{3} \right) + f(b) \right] \end{aligned}

where h=b−a3h = \dfrac{b − a}{3} is the step size.

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