Simpson 1/3

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 13\frac{1}{3} states that:

∫abf(x)  dx≈13h[f(a)+4f(a+b2)+f(b)]≈b−a6[f(a)+4f(a+b2)+f(b)]\begin{aligned} \int_a^b f(x)\;dx & \approx \frac{1}{3} h \left[ f(a) + 4f\left(\frac{a + b}{2} \right) + f(b) \right] \\ & \approx \frac{b - a}{6} \left[ f(a) + 4f\left(\frac{a+b}{2} \right) + f(b) \right] \end{aligned}

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