Question

use the trapezoidal rule, the midpoint rule, and simpsons rule to approximate the given integral with the specified value of n. (round your answers to six decimal places.) (int_{0}^{12}x^{3}sin(x)dx, n = 8) (a) the trapezoidal rule -1392.087125 good job. (b) the midpoint rule -1744.434609 nice job! (c) simpsons rule -1636.408033 good work.

Explanation:

Step1: Define the integral and parameters

The integral is $\int_{0}^{12}x^{3}\sin(x)dx$ and $n = 8$. First, find the width of each sub - interval $\Delta x=\frac{b - a}{n}$, where $a = 0$, $b = 12$ and $n=8$. So, $\Delta x=\frac{12-0}{8}=1.5$.

Step2: Trapezoidal rule

The trapezoidal rule formula is $T_n=\frac{\Delta x}{2}[f(x_0)+2\sum_{i = 1}^{n - 1}f(x_i)+f(x_n)]$.
Let $x_i=a + i\Delta x$. Calculate $f(x)=x^{3}\sin(x)$ at each $x_i$.
$x_0 = 0,f(x_0)=0$; $x_1=1.5,f(x_1)=(1.5)^{3}\sin(1.5)$; $x_2 = 3,f(x_2)=3^{3}\sin(3)$; $x_3=4.5,f(x_3)=(4.5)^{3}\sin(4.5)$; $x_4 = 6,f(x_4)=6^{3}\sin(6)$; $x_5=7.5,f(x_5)=(7.5)^{3}\sin(7.5)$; $x_6 = 9,f(x_6)=9^{3}\sin(9)$; $x_7=10.5,f(x_7)=(10.5)^{3}\sin(10.5)$; $x_8 = 12,f(x_8)=12^{3}\sin(12)$.
Then $T_8=\frac{1.5}{2}[0 + 2(f(x_1)+f(x_2)+f(x_3)+f(x_4)+f(x_5)+f(x_6)+f(x_7))+f(x_8)]\approx - 1392.087125$.

Step3: Mid - point rule

The mid - point rule formula is $M_n=\Delta x\sum_{i = 0}^{n - 1}f(\frac{x_i+x_{i + 1}}{2})$.
The mid - points are $m_0=\frac{0 + 1.5}{2}=0.75,m_1=\frac{1.5+3}{2}=2.25,m_2=\frac{3 + 4.5}{2}=3.75,m_3=\frac{4.5+6}{2}=5.25,m_4=\frac{6+7.5}{2}=6.75,m_5=\frac{7.5+9}{2}=8.25,m_6=\frac{9+10.5}{2}=9.75,m_7=\frac{10.5+12}{2}=11.25$.
Calculate $f(m_i)=m_i^{3}\sin(m_i)$ for each $m_i$ and sum them up: $M_8=1.5\sum_{i = 0}^{7}f(m_i)\approx - 1744.434609$.

Step4: Simpson's rule

The Simpson's rule formula is $S_n=\frac{\Delta x}{3}[f(x_0)+4\sum_{i = 1, i\text{ odd}}^{n - 1}f(x_i)+2\sum_{i = 2, i\text{ even}}^{n - 2}f(x_i)+f(x_n)]$.
Using the values of $x_i$ and $f(x)=x^{3}\sin(x)$ calculated before:
$S_8=\frac{1.5}{3}[0+4(f(x_1)+f(x_3)+f(x_5)+f(x_7))+2(f(x_2)+f(x_4)+f(x_6))+f(x_8)]\approx - 1636.408033$.

Answer:

(a) - 1392.087125
(b) - 1744.434609
(c) - 1636.408033