Answer
$$7.14$$
Work Step by Step
Given $$ f(x)=\sqrt{6 x+2}, \quad[1,3]$$
Since $n=6$, $\Delta x= \dfrac{b-a}{n}=\dfrac{1}{3}$ and
$$x_0= 1,\ x_1= 4/3,\ x_2= 5/3,\ x_3=2,\ x_4= 7/3,\ x_5= 8/3,\ x_6= 3$$
Then
\begin{align*}
L_{n}&=\left[f(x_0)+f(x_1)+.......+f(x_{n-1})\right]\Delta x\\
L_6&=\left[f(x_0)+f(x_1)+.......+f(x_{5})\right]\Delta x\\
&=\left[ f(1)+ f( 4/3)+ f( 5/3)+f( 2)+f( 7/3)+f( 8/3)\right]\frac{1}{3}\\
&= \left[ \sqrt{8}+\sqrt{10}+\sqrt{12}+\sqrt{14} +\sqrt{16}+\sqrt{18} \right]\frac{1}{3}\\
&=\approx 7.14
\end{align*}
