use the figures to calculate the left and right riemann sums for f on the given interval and the given value…

Explanation:

Step1: Calculate the width of sub - intervals

The interval is $[a,b]=[0,5]$ and $n = 5$. The width of each sub - interval $\Delta x=\frac{b - a}{n}=\frac{5-0}{5}=1$.

Step2: Determine the left - hand endpoints

The left - hand endpoints of the sub - intervals $x_i$ for $i = 0,1,2,3,4$ are $x_0=0,x_1 = 1,x_2=2,x_3=3,x_4=4$.

Step3: Evaluate the function at left - hand endpoints

$f(x)=x + 2$. So $f(x_0)=f(0)=0 + 2=2$, $f(x_1)=f(1)=1 + 2=3$, $f(x_2)=f(2)=2+2 = 4$, $f(x_3)=f(3)=3 + 2=5$, $f(x_4)=f(4)=4 + 2=6$.

Step4: Calculate the left Riemann sum

The left Riemann sum $L_n=\sum_{i = 0}^{n - 1}f(x_i)\Delta x$. Here, $L_5=f(x_0)\Delta x+f(x_1)\Delta x+f(x_2)\Delta x+f(x_3)\Delta x+f(x_4)\Delta x$. Since $\Delta x = 1$, $L_5=2\times1+3\times1+4\times1+5\times1+6\times1=2 + 3+4+5+6=20$.

Answer:

20