approximate the area under the curve graphed below from (x = 2) to (x = 6) using a left endpoint…

approximate the area under the curve graphed below from (x = 2) to (x = 6) using a left endpoint approximation with 4 subdivisions. (you will need to approximate the function values using the graph.)

approximate the area under the curve graphed below from (x = 2) to (x = 6) using a left endpoint approximation with 4 subdivisions. (you will need to approximate the function values using the graph.)

Answer

Explanation:

Step1: Calculate width of sub - intervals

The interval is from $a = 2$ to $b = 6$ and $n=4$. The width of each sub - interval $\Delta x=\frac{b - a}{n}=\frac{6 - 2}{4}=1$.

Step2: Identify left - endpoints

The left - endpoints of the 4 sub - intervals $[2,3]$, $[3,4]$, $[4,5]$, $[5,6]$ are $x_1 = 2$, $x_2 = 3$, $x_3 = 4$, $x_4 = 5$.

Step3: Estimate function values from graph

Let's assume from the graph that $f(2)\approx1.5$, $f(3)\approx2.5$, $f(4)\approx3.2$, $f(5)\approx3.7$.

Step4: Calculate left - endpoint approximation

The left - endpoint approximation $L_4=\sum_{i = 1}^{4}f(x_i)\Delta x$. Since $\Delta x = 1$, $L_4=f(2)\times1+f(3)\times1+f(4)\times1+f(5)\times1$. Substituting the values, we get $L_4=1.5 + 2.5+3.2 + 3.7=10.9$.

Answer:

$10.9$