Breaking down a numerical integration into two parts

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Luqman Saleem
Luqman Saleem 2024-3-6
评论： Luqman Saleem 2024-3-6
Ran in:
Sorry for this stupic question.
Suppose we have this integral
Of couse this can be solved analytically. But I still want to solve it numerically as:
dx = 0.01;
x = 1:dx:4;
fx = x.^2 + 5*x;
I = trapz(x,fx)
I = 58.5001
Now, we break down this integral into two part. Analytically, we write it as:
I am confused about the lower limit of the second integral when solving it numerically. Should it be 2 or . If we use 2, will not it mean that we are counting one point two times, once in and once in ?

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Torsten
Torsten 2024-3-6
I1 = dx/2 * f(x0) + dx * sum_{i=1}^{n-1} f(xi) + dx/2 * f(xn)
I2 = dx/2 * f(xn) + dx * sum_{i=n+1}^{n+m-1} f(xi) + dx/2 * f(x_{n+m})
->
I1 + I2 = dx/2 * f(x0) + dx * sum_{i=1}^{n+m-1} f(xi) + dx/2 * f(x_{n+m})
Thus since the endpoints in the trapezoidal rule are only counted half, all is fine.

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