Esercizi sull'Integrazione Numerica N° 3

$\int_{0}^{1} \frac{1}{1 + x^{2}} ⅆ x = {[a r c t g (x)]}_{0}^{1} = a r c t g (1) = \frac{π}{4} \to π = 4 \cdot \int_{0}^{1} \frac{1}{1 + x^{2}} ⅆ x$
Ampiezza del sottointervallo: ∆x= (b - a)/n= 1/10= 0.1
Metodo dei rettangoli $\int_{0}^{1} \frac{1}{1 + x^{2}} ⅆ x = \frac{1}{10} [f (0) + f (0.1) + f (0.2) + f (0.3) + f (0.4) + f (0.5) + f (0.6) + f (0.7) + f (0.8) + f (0.9)] =$ $= \frac{1}{10} (\frac{1}{1 + 0^{2}} + \frac{1}{1 + 0. 1^{2}} + \frac{1}{1 + 0. 2^{2}} + \frac{1}{1 + 0. 3^{2}} + \frac{1}{1 + 0. 4^{2}} + \frac{1}{1 + 0. 5^{2}} + \frac{1}{1 + 0. 6^{2}} + \frac{1}{1 + 0. 7^{2}} + \frac{1}{1 + 0. 8^{2}} + \frac{1}{1 + 0. 9^{2}}) = 0.809981497$ $π = 4 \cdot \int_{0}^{1} \frac{1}{1 + x^{2}} ⅆ x = 4 \cdot 0.809981497 = 3.239925988$
Metodo dei trapezi $\int_{0}^{1} \frac{1}{1 + x^{2}} ⅆ x = \frac{1}{10} [\frac{f (0) + f (1)}{2} + f (0.1) + f (0.2) + f (0.3) + f (0.4) + f (0.5) + f (0.6) + f (0.7) + f (0.8) + f (0.9)] =$ $= \frac{1}{10} [\frac{\frac{1}{1 + 0^{2}} + \frac{1}{1 + 1^{2}}}{2} + \frac{1}{1 + 0. 1^{2}} + \frac{1}{1 + {0.2}^{2}} + \frac{1}{1 + 0. 3^{2}} + \frac{1}{1 + {0.4}^{2}} + \frac{1}{1 + {0.5}^{2}} + \frac{1}{1 + {0.6}^{2}} + \frac{1}{1 + {0.7}^{2}} + \frac{1}{1 + {0.8}^{2}} + \frac{1}{1 + {0.9}^{2}}] = 0.7849814972$ $π = 4 \cdot \int_{0}^{1} \frac{1}{1 + x^{2}} ⅆ x = 4 \cdot 0.7849814972 = 3.139925989$
Metodo delle parabole $\int_{0}^{1} \frac{1}{1 + x^{2}} ⅆ x = \frac{1}{3} \cdot \frac{1}{10} {f (0) + f (1) + 2 \cdot [f (0.2) + f (0.4) + f (0.6) + f (0.8)] + 4 \cdot [f (0.1) + f (0.3) + f (0.5) + f (0.7) + f (0.9)]} =$ $= \frac{1}{30} [\frac{1}{1 + 0^{2}} + \frac{1}{1 + 1^{2}} + 2 \cdot (\frac{1}{1 + 0. 2^{2}} + \frac{1}{1 + 0. 4^{2}} + \frac{1}{1 + 0. 6^{2}} + \frac{1}{1 + 0. 8^{2}}) + 4 \cdot (\frac{1}{1 + 0. 1^{2}} + \frac{1}{1 + 0. 3^{2}} + \frac{1}{1 + 0. 5^{2}} + \frac{1}{1 + 0. 7^{2}} + \frac{1}{1 + 0. 9^{2}})] = 0.7853981535$ $π = 4 \cdot \int_{0}^{1} \frac{1}{1 + x^{2}} ⅆ x = 4 \cdot 0.7853981535 = 3.141592614$