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I was learning about the Trapezoid from Paul's Online Notes (a great resource!). He mentions that "all function evaluations, with the exceptions of the first and last, are multiplied by two."
What does this mean? How does this look practically?
This is the link to the page:
1 Answer
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The trapezoidal rule for numerical approximation of an integral is $$\int _a^b f(x)dx \approx $$
$$ T= ( h/2)[f(x_0)+2f(x_1) + 2f(x_2) +...+ 2f(x_{n-1}) + f(x_n)]$$
Where the interval $[a,b]$ is divided into $n$ sub-intervals of size $h=(b-a)/n$ with partition points of $$\{a=x_0, x_1, x_2,...,x_n=b\}$$
The reason for multiplying some $f(x_i)$ by $2$ is the $f(x_i)$ is shared by two adjacent trapezoids so it counts twice in finding the total areas of trapezoids.
