Chapter 5 - The Integral - 5.1 Approximating and Computing Area - Exercises - Page 235: 18

$$1.12$$

Given $$f(x)=x^{2}+x, \quad[-1,1]$$ Since $n=5$, then $\Delta x=\dfrac{1+1}{5}=0.4$ and $$x_0=-1,\ x_1=-0.6,\ x_2= -0.2,\ x_3= 0.2,\ x_4=0.6,\ x_5= 1 $$ Then \begin{align*} R_{n}&=\left[f(x_1)+f(x_2)+.......+f(x_{n })\right]\Delta x\\ R_5&=\left[f(x_1 )+f(x_2)+.......+f(x_{5})\right]\Delta x\\ &=\left[f(-0.6)+f(-0.2)+f(0.2)+f(0.6)+f(1)\right](0.4)\\ &=[-0.24-0.16+0.24+0.96+2 ]0.4\\ &=1.12 \end{align*}