Chapter 5 - The Integral - 5.1 Approximating and Computing Area - Exercises - Page 235: 20

$$3.99$$

Given $$ f(x)=\sqrt{x}, \quad[3,5]$$ Since $n= 4$, $\Delta x=\dfrac{1}{2}$ and $$ x_0= 3,\ x_1=3.5,\ x_2=4,\ x_3=4.5,\ x_4=5 $$ 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_4&= \left[f(3.25)+f(3.75)+f(4.25)+f(4.75) \right] \Delta x\\ &= [ \sqrt{3.25}+ \sqrt{3.75}+ \sqrt{4.25}+ \sqrt{4.75} ](0.5)\\ &=3.99 \end{align*}