Esercizi sull'Integrazione Numerica N° 4

$\int_{1}^{2} \frac{4}{x^{2} + 16} ⅆ x = \int_{1}^{2} \frac{1}{{(\frac{x}{4})}^{2} + 1} ⅆ x = \int_{\frac{1}{4}}^{\frac{1}{2}} \frac{1}{{(u)}^{2} + 1} ⅆ u = {[a r c t g (u)]}_{\frac{1}{4}}^{\frac{1}{2}} = a r c t g (\frac{1}{2}) - a r c t g (\frac{1}{4}) ≃ 0.218668945874$
Ampiezza del sottointervallo: ∆x= (b - a)/n= 1/10= 0.1
Metodo dei rettangoli $\int_{1}^{2} \frac{4}{x^{2} + 16} ⅆ x = \frac{1}{10} [f (1) + f (1.1) + f (1.2) + f (1.3) + f (1.4) + f (1.5) + f (1.6) + f (1.7) + f (1.8) + f (1.9)] =$ $= \frac{1}{10} [\frac{4}{1^{2} + 16} + \frac{4}{1. 1^{2} + 16} + \frac{4}{1. 2^{2} + 16} + \frac{4}{1. 3^{2} + 16} + \frac{4}{1. 4^{2} + 16} + \frac{4}{1. 5^{2} + 16} + \frac{4}{1. 6^{2} + 16} + \frac{4}{1. 7^{2} + 16} + \frac{4}{1. 8^{2} + 16} + \frac{4}{1. 9^{2} + 16}] = 0.2204233869$
Metodo dei trapezi $\int_{1}^{2} \frac{4}{x^{2} + 16} ⅆ x = \frac{1}{10} [\frac{f (1) + f (2)}{2} + f (1.1) + f (1.2) + f (1.3) + f (1.4) + f (1.5) + f (1.6) + f (1.7) + f (1.8) + f (1.9)] =$ $= \frac{1}{10} [\frac{\frac{4}{1^{2} + 16} + \frac{4}{2^{2} + 16}}{2} + \frac{4}{1. 1^{2} + 16} + \frac{4}{1. 2^{2} + 16} + \frac{4}{1. 3^{2} + 16} + \frac{4}{1. 4^{2} + 16} + \frac{4}{1. 5^{2} + 16} + \frac{4}{1. 6^{2} + 16} + \frac{4}{1. 7^{2} + 16} + \frac{4}{1. 8^{2} + 16} + \frac{4}{1. 9^{2} + 16}] = 0.2188575591$
Metodo delle parabole $\int_{1}^{2} \frac{4}{x^{2} + 16} ⅆ x = \frac{1}{30} {f (1) + f (2) + 2 [f (1.2) + f (1.4) + f (1.6) + f (1.8)] + 4 [f (1.1) + f (1.3) + f (1.5) + f (1.7) + f (1.9)]} =$ $= \frac{1}{30} {\frac{4}{1^{2} + 16} + \frac{4}{2^{2} + 16} + 2 [\frac{4}{1. 2^{2} + 16} + \frac{4}{1. 4^{2} + 16} + \frac{4}{1. 6^{2} + 16} + \frac{4}{1. 8^{2} + 16}] + 4 [\frac{4}{1. 1^{2} + 16} + \frac{4}{1. 3^{2} + 16} + \frac{4}{1. 5^{2} + 16} + \frac{4}{1. 7^{2} + 16} + \frac{4}{1. 9^{2} + 16}]} = 0.2186689443$