Chapter 5 - The Integral - 5.1 Approximating and Computing Area - Exercises - Page 235: 19

$$1.3$$

Given $$ f(x)=\ln x, \quad[1,3]$$ Since $n= 5$, $\Delta x=\frac{2}{5}= 0.4$ and $$ x_0= 1,\ x_1=1.4,\ x_2=1.8,\ x_3=2.2,\ x_4=2.6,\ x_5=3 $$ Then \begin{align*} M_n&=\left[f\left(\frac{x_{0}+x_{1}}{2}\right)+\cdots+f\left(\frac{x_{n-1}+x_{n}}{2}\right)\right] \Delta x\\ M_6&= \left[f(1.2)+f(1.6)+f(2)+f(2.4)+f(2.8) \right] \Delta x\\ &= [ \ln 1.2+\ln 1.6+\ln 2+\ln 2.4+\ln 2.8](0.4)\\ &=1.3 \end{align*}