Anno 2008 Suppletiva – 10 – Rettangoli di sinistra

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Tenuto conto che \displaystyle \frac{\pi}{6}=\int_0^{\frac{1}{2}}\frac{1}{\sqrt{1-x^2}}\, dx si calcoli un’approssimazione di π, utilizzando uno dei metodi d’integrazione numerica studiati.

Metodo dei rettangoli con altezze di sinistra

nhx_iy_iArea= \displaystyle h\cdot (y_0)
1\displaystyle \frac{1}{2}01= \displaystyle\frac{1}{2}\cdot (1)
= \displaystyle\frac{1}{2}
Pi greco= 6\cdot Area
= \displaystyle 6\cdot\left(\frac{1}{2}\right)
= 3,0
nhx_iy_iArea= \displaystyle h\cdot (y_0+y_1)
2\displaystyle \frac{1}{4}01= \displaystyle\frac{1}{4}\cdot\left(1+\frac{4\sqrt{15}}{15}}\right)
\displaystyle \frac{1}{4}\displaystyle \frac{4\sqrt{15}}{15}= \displaystyle\frac{1}{4}+\frac{\sqrt{15}}{15}}
Pi greco= 6\cdot Area
= \displaystyle 6\cdot\left(\frac{1}{4}+\frac{\sqrt{15}}{15}}\right)
= \displaystyle\frac{3}{2}+\frac{2\sqrt{15}}{5}}
= 3,049193
nhx_iy_iArea= \displaystyle h\cdot (y_0+y_1+y_2)
3\displaystyle \frac{1}{6}01= \displaystyle\frac{1}{6}\cdot\left(1+\frac{6\sqrt{35}}{35}+\frac{3\sqrt{2}}{4}\right)
\displaystyle \frac{1}{6}\displaystyle \frac{6\sqrt{35}}{35}= \displaystyle\frac{1}{6}+\frac{\sqrt{2}}{8}+\frac{\sqrt{35}}{35}\right)
\displaystyle \frac{1}{3}\displaystyle \frac{3\sqrt{2}}{4}Pi greco= 6\cdot Area
= \displaystyle 6\cdot\left(\frac{1}{6}+\frac{\sqrt{2}}{8}+\frac{\sqrt{35}}{35}\right)\right)
= \displaystyle 1+\frac{3\sqrt{2}}{4}+\frac{6\sqrt{35}}{35}
= 3,074845
nhx_iy_iArea= \displaystyle h\cdot (y_0+y_1+y_2+y_3)
4\displaystyle \frac{1}{8}01= \displaystyle \frac{1}{8}\cdot\left(1 + \frac{8\sqrt{7}}{21} + \frac{4\sqrt{15}}{15} + \frac{8\sqrt{55}}{55}\right)
\displaystyle \frac{1}{8}\displaystyle \frac{8\sqrt{7}}{21}= \displaystyle \frac{1}{8} + \frac{\sqrt{7}}{21} + \frac{\sqrt{15}}{30} + \frac{\sqrt{55}}{55}\right)
\displaystyle \frac{1}{4}\displaystyle \frac{4\sqrt{15}}{15}Pi greco= 6\cdot Area
\displaystyle \frac{3}{8}\displaystyle \frac{8\sqrt{55}}{55}= \displaystyle 6\cdot\left(\frac{1}{8} + \frac{\sqrt{7}}{21} + \frac{\sqrt{15}}{30} + \frac{\sqrt{55}}{55}\right)\right)
= \displaystyle \frac{3}{4} + \frac{2\sqrt{7}}{7} + \frac{\sqrt{15}}{5} + \frac{6\sqrt{55}}{55}
= 3,089566
nh…
100\displaystyle \frac{1}{200}Pi greco= 3,139282