cedimientos: e una aproximación a la integral usando la suma de Rein - \( \int_{0}^{4} x^{3}-3 x d x \quad n=16 \) y \( n= \) intervalo

Para aproximar la integral \( \int_{0}^{4} (x^{3} - 3x) \, dx \) usando la suma de Riemann con \( n = 16 \), seguiremos los siguientes pasos: ### Paso 1: Determinar el ancho de cada subintervalo El intervalo de integración es de \( 0 \) a \( 4 \). El ancho de cada subintervalo \( \Delta x \) se calcula como: \[ \Delta x = \frac{b - a}{n} = \frac{4 - 0}{16} = \frac{4}{16} = 0.25 \] ### Paso 2: Determinar los puntos de evaluación Los puntos de evaluación \( x_i \) se pueden calcular como: \[ x_i = a + i \Delta x \quad \text{para } i = 0, 1, 2, \ldots, n \] Por lo tanto, los puntos son: - \( x_0 = 0 + 0 \cdot 0.25 = 0 \) - \( x_1 = 0 + 1 \cdot 0.25 = 0.25 \) - \( x_2 = 0 + 2 \cdot 0.25 = 0.5 \) - \( x_3 = 0 + 3 \cdot 0.25 = 0.75 \) - \( x_4 = 0 + 4 \cdot 0.25 = 1 \) - \( x_5 = 0 + 5 \cdot 0.25 = 1.25 \) - \( x_6 = 0 + 6 \cdot 0.25 = 1.5 \) - \( x_7 = 0 + 7 \cdot 0.25 = 1.75 \) - \( x_8 = 0 + 8 \cdot 0.25 = 2 \) - \( x_9 = 0 + 9 \cdot 0.25 = 2.25 \) - \( x_{10} = 0 + 10 \cdot 0.25 = 2.5 \) - \( x_{11} = 0 + 11 \cdot 0.25 = 2.75 \) - \( x_{12} = 0 + 12 \cdot 0.25 = 3 \) - \( x_{13} = 0 + 13 \cdot 0.25 = 3.25 \) - \( x_{14} = 0 + 14 \cdot 0.25 = 3.5 \) - \( x_{15} = 0 + 15 \cdot 0.25 = 3.75 \) - \( x_{16} = 0 + 16 \cdot 0.25 = 4 \) ### Paso 3: Evaluar la función en los puntos La función a evaluar es \( f(x) = x^{3} - 3x \). Evaluamos la función en cada uno de los puntos \( x_i \): \[ f(x_i) = x_i^{3} - 3x_i \] ### Paso 4: Calcular la suma de Riemann La suma de Riemann se calcula como: \[ S_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \] ### Paso 5: Realizar los cálculos Ahora, procederemos a calcular \( f(x_i) \) para cada \( i \) y luego la suma de Riemann. Voy a realizar los cálculos necesarios. Calculate the value by following steps: - step0: Calculate: \(16\times 0.25\) - step1: Multiply the numbers: \(4\) Calculate or simplify the expression \( (0)^(3) - 3*(0) + (0.25)^(3) - 3*(0.25) + (0.5)^(3) - 3*(0.5) + (0.75)^(3) - 3*(0.75) + (1)^(3) - 3*(1) + (1.25)^(3) - 3*(1.25) + (1.5)^(3) - 3*(1.5) + (1.75)^(3) - 3*(1.75) + (2)^(3) - 3*(2) + (2.25)^(3) - 3*(2.25) + (2.5)^(3) - 3*(2.5) + (2.75)^(3) - 3*(2.75) + (3)^(3) - 3*(3) + (3.25)^(3) - 3*(3.25) + (3.5)^(3) - 3*(3.5) + (3.75)^(3) - 3*(3.75) \). Calculate the value by following steps: - step0: Calculate: \(0^{3}-3\times 0+0.25^{3}-3\times 0.25+0.5^{3}-3\times 0.5+0.75^{3}-3\times 0.75+1^{3}-3\times 1+1.25^{3}-3\times 1.25+1.5^{3}-3\times 1.5+1.75^{3}-3\times 1.75+2^{3}-3\times 2+2.25^{3}-3\times 2.25+2.5^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step1: Multiply: \(0^{3}+0+0.25^{3}-3\times 0.25+0.5^{3}-3\times 0.5+0.75^{3}-3\times 0.75+1^{3}-3\times 1+1.25^{3}-3\times 1.25+1.5^{3}-3\times 1.5+1.75^{3}-3\times 1.75+2^{3}-3\times 2+2.25^{3}-3\times 2.25+2.5^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step2: Calculate: \(0+0+0.25^{3}-3\times 0.25+0.5^{3}-3\times 0.5+0.75^{3}-3\times 0.75+1^{3}-3\times 1+1.25^{3}-3\times 1.25+1.5^{3}-3\times 1.5+1.75^{3}-3\times 1.75+2^{3}-3\times 2+2.25^{3}-3\times 2.25+2.5^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step3: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+0.5^{3}-3\times 0.5+0.75^{3}-3\times 0.75+1^{3}-3\times 1+1.25^{3}-3\times 1.25+1.5^{3}-3\times 1.5+1.75^{3}-3\times 1.75+2^{3}-3\times 2+2.25^{3}-3\times 2.25+2.5^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step4: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+0.75^{3}-3\times 0.75+1^{3}-3\times 1+1.25^{3}-3\times 1.25+1.5^{3}-3\times 1.5+1.75^{3}-3\times 1.75+2^{3}-3\times 2+2.25^{3}-3\times 2.25+2.5^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step5: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1^{3}-3\times 1+1.25^{3}-3\times 1.25+1.5^{3}-3\times 1.5+1.75^{3}-3\times 1.75+2^{3}-3\times 2+2.25^{3}-3\times 2.25+2.5^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step6: Evaluate the power: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+1.25^{3}-3\times 1.25+1.5^{3}-3\times 1.5+1.75^{3}-3\times 1.75+2^{3}-3\times 2+2.25^{3}-3\times 2.25+2.5^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step7: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+1.5^{3}-3\times 1.5+1.75^{3}-3\times 1.75+2^{3}-3\times 2+2.25^{3}-3\times 2.25+2.5^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step8: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+1.75^{3}-3\times 1.75+2^{3}-3\times 2+2.25^{3}-3\times 2.25+2.5^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step9: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+2.25^{3}-3\times 2.25+2.5^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step10: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+2.5^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step11: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+2.75^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step12: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+3.25^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step13: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+3.5^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step14: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+3.75^{3}-3\times 3.75\) - step15: Convert the expressions: \(0+0+\left(\frac{1}{4}\right)^{3}-3\times 0.25+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step16: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-3\times 0.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step17: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-3\times 0.75+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step18: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3\times 1+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step19: Multiply: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3\times 1.25+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step20: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-3\times 1.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step21: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-3\times 1.75+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step22: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-3\times 2+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step23: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-3\times 2.25+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step24: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-3\times 2.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step25: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-3\times 2.75+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step26: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-3\times 3+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step27: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-3\times 3.25+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step28: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-3\times 3.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step29: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-3\times 3.75\) - step30: Multiply the numbers: \(0+0+\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step31: Remove 0: \(\left(\frac{1}{4}\right)^{3}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step32: Evaluate the power: \(\frac{1}{64}-0.75+\left(\frac{1}{2}\right)^{3}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step33: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\left(\frac{3}{4}\right)^{3}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step34: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\left(\frac{5}{4}\right)^{3}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step35: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\frac{125}{64}-3.75+\left(\frac{3}{2}\right)^{3}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step36: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\frac{125}{64}-3.75+\frac{27}{8}-4.5+\left(\frac{7}{4}\right)^{3}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step37: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\frac{125}{64}-3.75+\frac{27}{8}-4.5+\frac{343}{64}-5.25+2^{3}-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step38: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\frac{125}{64}-3.75+\frac{27}{8}-4.5+\frac{343}{64}-5.25+8-6+\left(\frac{9}{4}\right)^{3}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step39: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\frac{125}{64}-3.75+\frac{27}{8}-4.5+\frac{343}{64}-5.25+8-6+\frac{729}{64}-6.75+\left(\frac{5}{2}\right)^{3}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step40: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\frac{125}{64}-3.75+\frac{27}{8}-4.5+\frac{343}{64}-5.25+8-6+\frac{729}{64}-6.75+\frac{125}{8}-7.5+\left(\frac{11}{4}\right)^{3}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step41: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\frac{125}{64}-3.75+\frac{27}{8}-4.5+\frac{343}{64}-5.25+8-6+\frac{729}{64}-6.75+\frac{125}{8}-7.5+\frac{1331}{64}-8.25+3^{3}-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step42: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\frac{125}{64}-3.75+\frac{27}{8}-4.5+\frac{343}{64}-5.25+8-6+\frac{729}{64}-6.75+\frac{125}{8}-7.5+\frac{1331}{64}-8.25+27-9+\left(\frac{13}{4}\right)^{3}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step43: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\frac{125}{64}-3.75+\frac{27}{8}-4.5+\frac{343}{64}-5.25+8-6+\frac{729}{64}-6.75+\frac{125}{8}-7.5+\frac{1331}{64}-8.25+27-9+\frac{2197}{64}-9.75+\left(\frac{7}{2}\right)^{3}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step44: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\frac{125}{64}-3.75+\frac{27}{8}-4.5+\frac{343}{64}-5.25+8-6+\frac{729}{64}-6.75+\frac{125}{8}-7.5+\frac{1331}{64}-8.25+27-9+\frac{2197}{64}-9.75+\frac{343}{8}-10.5+\left(\frac{15}{4}\right)^{3}-11.25\) - step45: Evaluate the power: \(\frac{1}{64}-0.75+\frac{1}{8}-1.5+\frac{27}{64}-2.25+1-3+\frac{125}{64}-3.75+\frac{27}{8}-4.5+\frac{343}{64}-5.25+8-6+\frac{729}{64}-6.75+\frac{125}{8}-7.5+\frac{1331}{64}-8.25+27-9+\frac{2197}{64}-9.75+\frac{343}{8}-10.5+\frac{3375}{64}-11.25\) - step46: Calculate: \(\frac{1}{64}-54+\frac{1}{8}+\frac{27}{64}+\frac{125}{64}+\frac{27}{8}+\frac{343}{64}+\frac{729}{64}+\frac{125}{8}+\frac{1331}{64}+\frac{2197}{64}+\frac{343}{8}+\frac{3375}{64}\) - step47: Reduce fractions to a common denominator: \(\frac{1}{64}-\frac{54\times 64}{64}+\frac{8}{8\times 8}+\frac{27}{64}+\frac{125}{64}+\frac{27\times 8}{8\times 8}+\frac{343}{64}+\frac{729}{64}+\frac{125\times 8}{8\times 8}+\frac{1331}{64}+\frac{2197}{64}+\frac{343\times 8}{8\times 8}+\frac{3375}{64}\) - step48: Multiply the numbers: \(\frac{1}{64}-\frac{54\times 64}{64}+\frac{8}{64}+\frac{27}{64}+\frac{125}{64}+\frac{27\times 8}{8\times 8}+\frac{343}{64}+\frac{729}{64}+\frac{125\times 8}{8\times 8}+\frac{1331}{64}+\frac{2197}{64}+\frac{343\times 8}{8\times 8}+\frac{3375}{64}\) - step49: Multiply the numbers: \(\frac{1}{64}-\frac{54\times 64}{64}+\frac{8}{64}+\frac{27}{64}+\frac{125}{64}+\frac{27\times 8}{64}+\frac{343}{64}+\frac{729}{64}+\frac{125\times 8}{8\times 8}+\frac{1331}{64}+\frac{2197}{64}+\frac{343\times 8}{8\times 8}+\frac{3375}{64}\) - step50: Multiply the numbers: \(\frac{1}{64}-\frac{54\times 64}{64}+\frac{8}{64}+\frac{27}{64}+\frac{125}{64}+\frac{27\times 8}{64}+\frac{343}{64}+\frac{729}{64}+\frac{125\times 8}{64}+\frac{1331}{64}+\frac{2197}{64}+\frac{343\times 8}{8\times 8}+\frac{3375}{64}\) - step51: Multiply the numbers: \(\frac{1}{64}-\frac{54\times 64}{64}+\frac{8}{64}+\frac{27}{64}+\frac{125}{64}+\frac{27\times 8}{64}+\frac{343}{64}+\frac{729}{64}+\frac{125\times 8}{64}+\frac{1331}{64}+\frac{2197}{64}+\frac{343\times 8}{64}+\frac{3375}{64}\) - step52: Transform the expression: \(\frac{1-54\times 64+8+27+125+27\times 8+343+729+125\times 8+1331+2197+343\times 8+3375}{64}\) - step53: Multiply the numbers: \(\frac{1-3456+8+27+125+27\times 8+343+729+125\times 8+1331+2197+343\times 8+3375}{64}\) - step54: Multiply the numbers: \(\frac{1-3456+8+27+125+216+343+729+125\times 8+1331+2197+343\times 8+3375}{64}\) - step55: Multiply the numbers: \(\frac{1-3456+8+27+125+216+343+729+1000+1331+2197+343\times 8+3375}{64}\) - step56: Multiply the numbers: \(\frac{1-3456+8+27+125+216+343+729+1000+1331+2197+2744+3375}{64}\) - step57: Calculate: \(\frac{8640}{64}\) - step58: Reduce the numbers: \(\frac{135}{1}\) - step59: Calculate: \(135\) ### Paso 6: Resultados de los cálculos 1. La suma de las evaluaciones de la función \( f(x_i) \) en los puntos \( x_i \) es \( 135 \). 2. El ancho de cada subintervalo \( \Delta x \) es \( 0.25 \). ### Paso 7: Calcular la suma de Riemann Ahora, calculamos la suma de Riemann: \[ S_n = \sum_{i=0}^{n-1} f(x_i) \Delta x = 135 \cdot 0.25 = 33.75 \] ### Resultado Final La aproximación de la integral \( \int_{0}^{4} (x^{3} - 3x) \, dx \) usando la suma de Riemann con \( n = 16 \) es: \[ \boxed{33.75} \]