Trapezoidal Rule

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 trapezoidal rule gives the following approximation:

Ii=∫xixi+1f(x)  dx≈12[f(xi)+f(xi+1)](xi+1−xi)I_i = \int_{x_i}^{x_{i + 1}} f(x)\;dx \approx \frac{1}{2} \left[ f(x_i) + f(x_{i + 1}) \right](x_{i + 1} - x_i)

By making h=b−anh = \dfrac{b - a}{n}, we get:

∫abf(x)  dx=∑i=0n−1Ii≈h2∑i=0n−1[f(xi)+f(xi+1)]∫abf(x)  dx≈h[f(a)+f(b)2+∑i=1n−1f(xi)]\begin{aligned} \int_a^b f(x)\;dx & = \sum_{i = 0}^{n - 1} I_i \approx \frac{h}{2} \sum_{i = 0}^{n - 1} [f(x_i) + f(x_{i+ 1})] \\ \int_a^b f(x)\;dx & \approx h \left[\dfrac{f(a) + f(b)}{2} + \sum_{i = 1}^{n - 1} f(x_i) \right] \end{aligned}

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