given the graph of the function f(x) below, use a left riemann sum with 6 rectangles to approximate the…

Explanation:

Step1: Calculate the width of each rectangle

The interval is from $a = 3$ to $b = 6$, and $n=6$. The width $\Delta x=\frac{b - a}{n}=\frac{6 - 3}{6}=0.5$.

Step2: Determine the left - hand endpoints

The left - hand endpoints of the 6 sub - intervals are $x_0 = 3$, $x_1=3 + 0.5=3.5$, $x_2=3.5 + 0.5 = 4$, $x_3=4+0.5 = 4.5$, $x_4=4.5 + 0.5=5$, $x_5=5 + 0.5 = 5.5$.

Step3: Estimate the function values at the left - hand endpoints

From the graph, $f(x_0)\approx6$, $f(x_1)\approx4$, $f(x_2)\approx3$, $f(x_3)\approx3$, $f(x_4)\approx3$, $f(x_5)\approx1$.

Step4: Calculate the left Riemann sum

The left Riemann sum $L_6=\sum_{i = 0}^{5}f(x_i)\Delta x=\Delta x\left[f(x_0)+f(x_1)+f(x_2)+f(x_3)+f(x_4)+f(x_5)\right]$. Substitute $\Delta x = 0.5$ and the function values: $L_6=0.5\times(6 + 4+3+3+3+1)=0.5\times20 = 10$.

Answer:

$10$