Aproxima $$$\int\limits_{-1}^{1} x e^{x}\, dx$$$ con $$$n = 5$$$ usando la regla trapezoidal
Calculadora relacionada: Calculadora de regla trapezoidal para una tabla
Tu aportación
Aproximar la integral $$$\int\limits_{-1}^{1} x e^{x}\, dx$$$ con $$$n = 5$$$ usando la regla trapezoidal.
Solución
La regla trapezoidal usa trapezoides para aproximar el área:
$$$\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \frac{\Delta x}{2} \left(f{\left(x_{0} \right)} + 2 f{\left(x_{1} \right)} + 2 f{\left(x_{2} \right)} + 2 f{\left(x_{3} \right)}+\dots+2 f{\left(x_{n-2} \right)} + 2 f{\left(x_{n-1} \right)} + f{\left(x_{n} \right)}\right)$$$
donde $$$\Delta x = \frac{b - a}{n}$$$.
Tenemos que $$$f{\left(x \right)} = x e^{x}$$$, $$$a = -1$$$, $$$b = 1$$$ y $$$n = 5$$$.
Por lo tanto, $$$\Delta x = \frac{1 - \left(-1\right)}{5} = \frac{2}{5}$$$.
Divida el intervalo $$$\left[-1, 1\right]$$$ en $$$n = 5$$$ subintervalos de longitud $$$\Delta x = \frac{2}{5}$$$ con los siguientes puntos finales: $$$a = -1$$$, $$$- \frac{3}{5}$$$, $$$- \frac{1}{5}$$$, $$$\frac{1}{5}$$$, $$$\frac{3}{5}$$$, $$$1 = b$$$.
Ahora, simplemente evalúe la función en estos puntos finales.
$$$f{\left(x_{0} \right)} = f{\left(-1 \right)} = - \frac{1}{e}\approx -0.367879441171442$$$
$$$2 f{\left(x_{1} \right)} = 2 f{\left(- \frac{3}{5} \right)} = - \frac{6}{5 e^{\frac{3}{5}}}\approx -0.658573963312832$$$
$$$2 f{\left(x_{2} \right)} = 2 f{\left(- \frac{1}{5} \right)} = - \frac{2}{5 e^{\frac{1}{5}}}\approx -0.327492301231193$$$
$$$2 f{\left(x_{3} \right)} = 2 f{\left(\frac{1}{5} \right)} = \frac{2 e^{\frac{1}{5}}}{5}\approx 0.488561103264068$$$
$$$2 f{\left(x_{4} \right)} = 2 f{\left(\frac{3}{5} \right)} = \frac{6 e^{\frac{3}{5}}}{5}\approx 2.186542560468611$$$
$$$f{\left(x_{5} \right)} = f{\left(1 \right)} = e\approx 2.718281828459045$$$
Finalmente, simplemente sume los valores anteriores y multiplíquelos por $$$\frac{\Delta x}{2} = \frac{1}{5}$$$: $$$\frac{1}{5} \left(-0.367879441171442 - 0.658573963312832 - 0.327492301231193 + 0.488561103264068 + 2.186542560468611 + 2.718281828459045\right) = 0.807887957295251.$$$
Respuesta
$$$\int\limits_{-1}^{1} x e^{x}\, dx\approx 0.807887957295251$$$A