Science:Math Exam Resources/Courses/MATH101/April 2016/Question 09 (a)/Solution 1

Since $a=0$ and $b=2$, we have $\mathrm{\Delta }x=\frac{2-0}{6}=\frac{1}{3}$, and so

$x_0=0$, $x_1=\frac{1}{3}$, $x_2=\frac{2}{3}$, $x_3=1$, $x_4=\frac{4}{3}$, $x_5=\frac{5}{3}$, and $x_6=2$.

Since Simpson’s Rule with $n=6$ in general is

$\frac{\mathrm{\Delta }x}{3}(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+4f(x_5)+f(x_6)),$

with $f(x)=(x-3{)}^5$, the desired approximation is

$\frac{1/3}{3}((-3{)}^5+4(\frac{1}{3}-3{)}^5+2(\frac{2}{3}-3{)}^5+4(-2{)}^5+2(\frac{4}{3}-3{)}^5+4(\frac{5}{3}-3{)}^5+(-1{)}^5).$