1 [5 pontos] Calcule a integral abaixo pelo Método de Gauss-Legendre com n=5.
\int_{0}^{0.8} \cos \left(\frac{\pi x^{2}}{2}\right) d x

Passo 1

O enunciado pede a aproximação da integral
\int_{0}^{0.8}\cos\!\bigl(\tfrac{\pi x^{2}}{2}\bigr)\,dx
pelo Método de Gauss–Legendre com n=5. Vamos usar as fórmulas padrão de Gauss–Legendre em [-1,1] e o mapeamento linear para o intervalo [0,0.8].

Passo 2

Para n=5 os pontos t_i e pesos w_i em [-1,1] são:
\begin{aligned} &t_{1}=0,\quad w_{1}=0.568889,\\\ &t_{2}=\pm0.538469,\quad w_{2}=0.478629,\\\ &t_{3}=\pm0.906180,\quad w_{3}=0.236927. \end{aligned}

Passo 3

O mapeamento de t\in[-1,1] para x\in[0,0.8] é
x = \frac{0.8-0}{2}\,t + \frac{0.8+0}{2} = 0.4\,t + 0.4.
Assim a integral fica
\int_{0}^{0.8}\cos\!\bigl(\tfrac{\pi x^{2}}{2}\bigr)\,dx =\frac{0.8-0}{2}\sum_{i=1}^{5}w_{i}\, \cos\!\Bigl(\tfrac{\pi\,(x_{i})^{2}}{2}\Bigr),
onde x_{i}=0.4\,t_{i}+0.4.

Passo 4

Calculamos x_{i} e \cos\!\bigl(\tfrac{\pi x_{i}^{2}}{2}\bigr) para cada t_{i}:
Para t_{1}=0:
x_{1}=0.4,\quad \cos\!\bigl(\tfrac{\pi\,(0.4)^{2}}{2}\bigr) =\cos(0.08\pi)\approx0.96844.
Para t_{2}=0.538469:
x_{2}=0.4\cdot0.538469+0.4\approx0.615388,\quad \cos\!\bigl(\tfrac{\pi\,(0.615388)^{2}}{2}\bigr) \approx0.82750.
Para t_{3}=-0.538469:
x_{3}=0.4\cdot(-0.538469)+0.4\approx0.184612,\quad \cos\!\bigl(\tfrac{\pi\,(0.184612)^{2}}{2}\bigr) \approx0.99856.
Para t_{4}=0.906180:
x_{4}=0.4\cdot0.906180+0.4\approx0.762472,\quad \cos\!\bigl(\tfrac{\pi\,(0.762472)^{2}}{2}\bigr) \approx0.61380.
Para t_{5}=-0.906180:
x_{5}=0.4\cdot(-0.906180)+0.4\approx0.037528,\quad \cos\!\bigl(\tfrac{\pi\,(0.037528)^{2}}{2}\bigr) \approx0.999998.

Passo 5

Agora montamos a soma pesada e aplicamos o fator \tfrac{0.8}{2}=0.4:
\begin{aligned} S &= w_{1}f(x_{1})+w_{2}f(x_{2})+w_{2}f(x_{3}) +w_{3}f(x_{4})+w_{3}f(x_{5})\\\ &=0.568889\cdot0.96844 +0.478629\cdot0.82750 +0.478629\cdot0.99856\\\ &\quad +0.236927\cdot0.61380 +0.236927\cdot0.999998 \approx1.808376. \end{aligned}
Portanto
\int_{0}^{0.8}\cos\!\bigl(\tfrac{\pi x^{2}}{2}\bigr)\,dx \approx0.4\times1.808376=0.72335.

Resposta

0.72335

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