$Iff$ has a continuous second derivative $ona,b,$ then the error $Ein$ approximating ${\displaystyle \underset{a}{\overset{b}{}}}f(x)dxby$ the

Trapezoidal Rule $is$

$|E|\frac{(b-a{)}^3}{12n^2}[max|f^{\text{'}\text{'}}(x)|],axb$

Moreover, $iff$ has a continuous fourth derivative $ona,b,$ then the error $Ein$ approximating ${\displaystyle \underset{a}{\overset{b}{}}}f(x)dxby$

Simpson's Rule $is$

$|E|\frac{(b-a{)}^5}{180n^4}[max|f^{(4)}(x)|],axb$

Use these $to$ find the minimum integer $n$ such that the error $in$ the approximation $of$ the definite integral $is$

less than $or$ equal $to0\mathrm{.}00001$ using the Trapezoidal Rule and Simpson's Rule.

${\displaystyle \underset{0}{\overset{8}{}}}\frac{1}{1+x}dx$