Integração numérica: Trapezoidal e Simpson

DanielFerreira escreveu: $\\ \text{Considere a integral definida abaixo:} \\\\ \mathrm{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx} \\\\\\ \text{(a) Calcule esta integral utilizando o \underline{m\'etodo dos trap\'ezios}, para n = 4 aplica\c{c}\~oes.}$

$\\ \mathsf{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx \approx A_1 + A_2 + A_3 + A_4} \\\\\\ \mathsf{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx \approx \frac{w}{2} \cdot \left [ f(0) + 2 \cdot f \left ( \frac{1}{4} \right ) + 2 \cdot f \left ( \frac{1}{2} \right ) + 2 \cdot f\left ( \frac{3}{4} \right ) + f \left ( 1 \right ) \right ]} \\\\\\ \mathsf{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx \approx \frac{\frac{1}{4}}{2} \cdot \left ( 0 + 2 \cdot 0,25770 + 2 \cdot 0,55902 + 2 \cdot 0,9375 + 1,41421 \right )}$

$\\ \mathsf{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx \approx \frac{1}{8} \cdot 4,92265} \\\\\\ \boxed{\boxed{\mathsf{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx \approx 0,61533}}}$

DanielFerreira escreveu: $\\ \text{Considere a integral definida abaixo:} \\\\ \mathrm{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx} \\\\\\ \text{(b) Calcule esta integral utilizando o \underline{m\'etodo de Simpson}, para n = 4 aplica\c{c}\~oes.}$

$\\ \mathsf{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx \approx S_1 + S_2 + S_3 + S_4} \\\\\\ \mathsf{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx \approx \frac{w}{3} \cdot \left [ f \left ( 0 \right ) + 4 \cdot f \left ( \frac{1}{8} \right ) + f \left ( \frac{1}{4} \right )\right ] + \frac{w}{3} \cdot \left [ f \left ( \frac{1}{4} \right ) + 4 \cdot f \left ( \frac{3}{8} \right ) + f \left ( \frac{1}{2} \right )\right ] +} \\\\\\ \mathsf{\qquad \qquad \qquad \qquad \frac{w}{3} \cdot \left [ f \left ( \frac{1}{2} \right ) + 4 \cdot f \left ( \frac{5}{8} \right ) + f \left ( \frac{3}{4} \right )\right ] + \frac{w}{3} \cdot \left [ f \left ( \frac{3}{4} \right ) + 4 \cdot f \left ( \frac{7}{8} \right ) + f \left ( 1 \right )\right ]}$

$\\ \mathsf{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx\approx \frac{1/8}{3} \cdot \left \{ f(0) + f(1) + 2 \cdot \left [ f \left ( \frac{1}{2} \right ) + f \left ( \frac{1}{4} \right ) + f \left ( \frac{3}{4} \right ) \right ] \right \} +} \\\\\\ \mathsf{\qquad \qquad \qquad \qquad \frac{1/8}{3} \cdot \left \{ 4 \cdot \left [ f \left ( \frac{1}{8} \right ) + f \left ( \frac{3}{8} \right ) + f \left ( \frac{5}{8} \right ) + f \left ( \frac{7}{8} \right )\right ] \right \}} \\\\\\ \mathsf{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx\approx \frac{1}{24} \cdot \left ( 0 + 1,41421 + 2 \cdot 1,75422 + 4 \cdot 2,42618 \right )}$

$\\ \mathsf{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx\approx \frac{1}{24} \cdot 14,62737} \\\\\\ \boxed{\boxed{\mathsf{\int_{0}^{1} x \sqrt{x^2 + 1} \ dx\approx 0,60947}}}$

Integração numérica: Trapezoidal e Simpson

Integração numérica: Trapezoidal e Simpson

Re: Integração numérica: Trapezoidal e Simpson

Re: Integração numérica: Trapezoidal e Simpson

Re: Integração numérica: Trapezoidal e Simpson