Regla Simpson
#FIG 3.2
Landscape
Center
Metric
A4
100.00
Single
-2
1200 2
1 4 0 1 0 0 50 0 20 4.000 1 0.0000 730 1155 23 23 708 1155 753 1155
1 4 0 1 0 0 50 0 20 4.000 1 0.0000 1260 785 23 23 1260 763 1260 808
1 4 0 1 0 0 50 0 20 4.000 1 0.0000 1805 1117 23 23 1805 1095 1805 1140
2 1 0 1 0 7 50 0 -1 4.000 0 0 -1 0 0 2
360 1980 2925 1980
2 1 1 1 8 0 50 0 -1 4.000 0 0 -1 0 0 2
720 1170 720 1980
2 1 1 1 8 0 50 0 -1 4.000 0 0 -1 0 0 2
1800 1125 1800 1980
2 1 1 1 8 0 50 0 -1 4.000 0 0 -1 0 0 2
1260 810 1260 1980
3 0 0 1 1 7 50 0 -1 4.000 0 0 0 8
585 1260 810 1080 1215 765 1530 720 1665 810 1710 990
1845 1215 2250 1260
0.000 1.000 1.000 1.000 1.000 1.000 1.000 0.000
3 0 0 1 4 0 50 0 -1 4.000 0 0 0 4
720 1170 1080 765 1530 765 1800 1125
0.000 1.000 1.000 0.000
3 0 0 1 23 0 50 0 -1 4.000 0 1 0 4
0 0 1.00 90.00 90.00
135 810 450 765 765 810 945 855
0.000 1.000 1.000 0.000
3 0 0 1 23 0 50 0 -1 4.000 0 1 0 3
0 0 1.00 90.00 90.00
2115 765 1890 720 1710 765
0.000 1.000 0.000
4 0 0 50 0 0 12 0.0000 4 135 90 1215 2160 2\001
4 0 0 50 0 0 12 0.0000 4 135 90 585 2160 1\001
4 0 0 50 0 0 12 0.0000 4 135 90 1665 2160 3\001
4 0 0 50 0 0 12 0.0000 4 135 90 2160 810 5\001
4 0 0 50 0 0 12 0.0000 4 135 90 2385 1170 6\001
4 0 0 50 0 0 12 0.0000 4 135 90 90 675 4\001
Finnum fleygboga í gegnum punktana $(x_{i-1}, y_{i-1}), (x_i, y_i), (x_{i+1}, y_{i+1})$
Nálgunargildi:
\begin{eqnarray*}
\int_a^b f(x)\,dx &\approx \frac{h}{3} \left( y_0 + 4y_1 + y_2 + y_2 + 4 y_3 + y_4 + y_4 + 4y_5 + y_6 + \cdots + y_{n-2} + 4y_{n-1} + y_n\right)\\
&= \frac{h}{3}\left( y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + \cdots + 2y_{n-2} + 4y_n + y_n \right)
\end{eqnarray*}
Details
Finnum fleygboga í gegnum punktana $(x_{i-1}, y_{i-1}), (x_i, y_i), (x_{i+1}, y_{i+1})$\\
\begin{eqnarray*}
\int_{-h}^h (Ax^2 + Bx + C)\,dx &= 2\int_0^h(Ax^2 + C)\,dx\\
&= 2A\frac{h^3}{3} + 2C\cdot h
\end{eqnarray*}
$y = Ax^2 + Bx + C$ gengur í gegnum $(-h,y_0), (0,y_1), (h,y_2)$, \quad $\Longrightarrow$
$$
y_0 = Ah^2 - Bh + C, \quad y_1 = C, \quad y_2 = Ah^2 + Bh + C
$$
$\Longrightarrow \qquad C = y_1$
$$
\begin{cases}
A h^2 - Bh = y_0 - y_1\\
Ah^2 + Bh = y_2 - y_1
\end{cases} \quad \Rightarrow \quad 2Ah^2 = y_0 - 2y_1 + y_2
$$
$\Longrightarrow$ \qquad Flatarmál undir $y = Ax^2 + Bx + C$ milli $x = -h$ og $x=h$ er:
\begin{eqnarray*}
\frac{h}{3} \left( 2Ah^2 + 6C\right) &= \frac{h}{3}\left( y_0 - 2y_1 + y_2 + 6y_1 \right)\\
&= \frac{h}{3} \left( y_0 + 4y_1 + y_2 \right)
\end{eqnarray*}
Beitum þessari formúlu á samliggjandi pör svæða:\\
Nálgunargildi:
\begin{eqnarray*}
\int_a^b f(x)\,dx &\approx \frac{h}{3} \left( y_0 + 4y_1 + y_2 + y_2 + 4 y_3 + y_4 + y_4 + 4y_5 + y_6 + \cdots + y_{n-2} + 4y_{n-1} + y_n\right)\\
&= \frac{h}{3}\left( y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + \cdots + 2y_{n-2} + 4y_n + y_n \right)
\end{eqnarray*}
\underline{Ath:}
$y_i = f(x_i), \qquad x_i = a + i\cdot h$\\
\underline{Ath:}
$n$ verður að vera slétt tala.