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^{2}sin(x)dx), (n = 8) (a) the trapezoidal rule (b) the midpoint rule (c) simpsons rule

Explanation:

Step1: Recall the formulas and calculate $\Delta x$

We have $a = 0$, $b=12$, $n = 8$. The width of each sub - interval $\Delta x=\frac{b - a}{n}=\frac{12-0}{8}=1.5$.

Step2: (a) Trapezoidal rule

The trapezoidal rule formula is $T_n=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+2f(x_2)+\cdots+2f(x_{n - 1})+f(x_n)]$.
The sub - endpoints are $x_i=a + i\Delta x$, $i = 0,1,\cdots,8$.
$x_0 = 0$, $x_1=1.5$, $x_2 = 3$, $\cdots$, $x_8=12$.
$T_8=\frac{1.5}{2}[f(0)+2f(1.5)+2f(3)+2f(4.5)+2f(6)+2f(7.5)+2f(9)+2f(10.5)+f(12)]$
$=\frac{1.5}{2}[0^2\sin(0)+2\times(1.5)^2\sin(1.5)+2\times3^2\sin(3)+2\times4.5^2\sin(4.5)+2\times6^2\sin(6)+2\times7.5^2\sin(7.5)+2\times9^2\sin(9)+2\times10.5^2\sin(10.5)+12^2\sin(12)]$
Using a calculator, $T_8\approx - 237.095197$.

Step3: (b) Midpoint rule

The mid - points of the sub - intervals are $m_i=x_{i-\frac{1}{2}}=a+(i - 0.5)\Delta x$, $i = 1,\cdots,8$.
$m_1=0.75$, $m_2 = 2.25$, $m_3=3.75$, $m_4 = 5.25$, $m_5=6.75$, $m_6 = 8.25$, $m_7=9.75$, $m_8 = 11.25$.
The midpoint rule formula is $M_n=\Delta x[f(m_1)+f(m_2)+\cdots+f(m_n)]$.
$M_8=1.5[(0.75)^2\sin(0.75)+(2.25)^2\sin(2.25)+(3.75)^2\sin(3.75)+(5.25)^2\sin(5.25)+(6.75)^2\sin(6.75)+(8.25)^2\sin(8.25)+(9.75)^2\sin(9.75)+(11.25)^2\sin(11.25)]$
Using a calculator, $M_8\approx - 268.134747$.

Step4: (c) Simpson's rule

Simpson's rule formula is $S_n=\frac{\Delta x}{3}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\cdots+2f(x_{n - 2})+4f(x_{n - 1})+f(x_n)]$ (where $n$ is even).
$S_8=\frac{1.5}{3}[f(0)+4f(1.5)+2f(3)+4f(4.5)+2f(6)+4f(7.5)+2f(9)+4f(10.5)+f(12)]$
$=\frac{1.5}{3}[0^2\sin(0)+4\times(1.5)^2\sin(1.5)+2\times3^2\sin(3)+4\times4.5^2\sin(4.5)+2\times6^2\sin(6)+4\times7.5^2\sin(7.5)+2\times9^2\sin(9)+4\times10.5^2\sin(10.5)+12^2\sin(12)]$
Using a calculator, $S_8\approx - 245.917027$.

Answer:

(a) $-237.095197$
(b) $-268.134747$
(c) $-245.917027$