Chapter 5 - The Integral - 5.1 Approximating and Computing Area - Exercises - Page 235: 22

$$12.125$$

Given $$ f(x)=x^{2}+3|x|, \quad[-2,1]$$ Since $n=6$, $\Delta x= \dfrac{b-a}{n}=\dfrac{1}{2}$ and $$x_0=-2,\ x_1= -1.5,\ x_2= -1,\ x_3=-0.5,\ x_4= 0,\ x_5= 0.5,\ x_6=1$$ 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(-2)+ f( -1.5)+ f( -1)+f( -0.5)+f( 0)+f( 0.5)\right]\frac{1}{3}\\ &= \left[ 10+6.75+4+1.75+1.75 \right](0.5)\\ &=12.125 \end{align*}