Chapter 5 - The Integral - 5.1 Approximating and Computing Area - Exercises - Page 235: 15

$$\frac{16}{3}$$

Given $$f(x)= 7-x,\ \ \ \ [ 3,5] $$ Since $n=3$, then $\Delta x=\dfrac{5-3}{3}=\dfrac{2}{3}$ and \begin{align*} R_{n}&=\left[f(x_1)+f(x_2)+.......+f(x_{n })\right]\Delta x\\ R_3&=\left[f(11/3 )+f(13/3) +f(5)\right]\Delta x\\ &=\left[ \frac{10}{3}+\frac{8}{3}+2\right]\dfrac{2}{3}\\ &= \frac{16}{3} \end{align*}