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Signed-off-by: David Rotermund <54365609+davrot@users.noreply.github.com>
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@ -38,7 +38,7 @@ Figure 7.2.: Scheme of the midpoint rule.
The integral is then replaced by the Riemann sum:
$I \approx I_M = \sum_{i=1}^N f(x_i)\Delta x$
$$I \approx I_M = \sum_{i=1}^N f(x_i)\Delta x$$
To illustrate the accuracy of this approximation, consider the following integral of the Gaussian bell-curve
@ -79,7 +79,9 @@ Insertion yields
$I_i = f(x_i)\Delta x+ \left[f'(x_i+\Delta x/2)- f'(x_i-\Delta x/2)\right]\frac{\Delta x^2}{24}+{\cal O}(\Delta x^5) \, . $
Taking the sum $I = \sum_{i=1}^N I_i$, many terms will vanish, thus
Taking the sum
$$I = \sum_{i=1}^N I_i$$
, many terms will vanish, thus
$I-I_M = \frac{\Delta x^2}{24}\left[f'(b)-f'(a)\right]+{\cal O}(\Delta x^4) \ .$
@ -101,7 +103,7 @@ $I_1 = \int_{x_1}^{x_2}f(x)dx \approx f(x_1)\Delta x+\frac{1}{2}[f(x_2)-f(x_1)]\
It follows the 'compound' trapezoid rule:
$I \approx I_T =\frac{1}{2}\left[f(x_1)+f(x_N)\right]\Delta x + \sum_{i=2}^{N-1} f(x_i)\Delta x \, .$
$$I \approx I_T =\frac{1}{2}\left[f(x_1)+f(x_N)\right]\Delta x + \sum_{i=2}^{N-1} f(x_i)\Delta x \, .$$
Note that the trapezoid rule contains the boundaries $a$ and $b$ of the interval. This is referred to as a closed formula. In contrast, the midpoint rule does not contain the boundaries of the interval (open formula). The latter is an advantage, in case that integrable singularities exist at $x=a$ or $x=b$.