Answer
$3.3829$
Work Step by Step
\[\begin{align}
& y=\ln x\text{ and }y={{x}^{3}}-5{{x}^{2}}+6x-1 \\
& \text{Find the intersection points let }y=y \\
& \ln x={{x}^{3}}-5{{x}^{2}}+6x-1 \\
& \text{Using a graphing calculator we obtain:} \\
& {{x}_{1}}=1.402,\text{ }{{x}_{2}}=3.448 \\
& \ln x\ge {{x}^{3}}-5{{x}^{2}}+6x-1\text{ on the interval }\left( 1.402,3.448 \right) \\
& \text{The area between the curves is:} \\
& A=\int_{1.402}^{3.448}{\left[ \ln x-\left( {{x}^{3}}-5{{x}^{2}}+6x-1 \right) \right]}dx \\
& \text{Integrating by a calculator we obtain} \\
& A\approx 3.3829 \\
\end{align}\]
