Question

Use Simpson's Rule to estimate  ∫_0^2▒1/8 e^(x^2 ) dx with a maximum error of 0.1

Answer 1

$Given, f(x)=\frac{e^{x^2}}{8}.\newline Exact\space value, \int_0^2\frac{e^{x^2}}{8}dx=2.0\space (2 \text{one decimal place})\newline \text{By using simpson's rule, the approximate value is} S_4\newline S_4=\frac{\Delta x}{3}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+f(x_4)]\newline where\space x_0=0, x_1=0.5, x_2=1,x_3=1.5, x_4=2, n=4, \Delta x=\frac{b-a}{n}=\frac{2-0}{4}=0.5\newline S_4=\frac{1}{3}[\frac{1}{8}+4(\frac{e^{0.5^2} }{8})+2(\frac{e}{8})+4(\frac{e^{1.5^2}}{8})+ (\frac{e^4}{8})]\newline =2.1 \space (2 \text{one decimal place}) \newline Error\leq 0.1\newline |exact-approx|=|2.0566-2.1692|\leq 0.1$