4. use the left endpoint approximation, with three intervals, to estimate the value of ∫-2 to 4(1 + x²)dx a…

Explanation:

Step1: Determine the width of intervals

The interval is from $a = - 2$ to $b = 4$. We want $n = 3$ intervals. The width of each interval $\Delta x=\frac{b - a}{n}=\frac{4-(-2)}{3}=\frac{6}{3}=2$.

Step2: Identify the left - endpoints

The left - endpoints of the three intervals $[-2,0]$, $[0,2]$, $[2,4]$ are $x_1=-2$, $x_2 = 0$, $x_3=2$.

Step3: Evaluate the function at left - endpoints

The function is $f(x)=1 + x^{2}$. Then $f(x_1)=f(-2)=1+(-2)^{2}=1 + 4 = 5$, $f(x_2)=f(0)=1+0^{2}=1$, $f(x_3)=f(2)=1+2^{2}=1 + 4 = 5$.

Step4: Calculate the left - endpoint approximation

The left - endpoint approximation $L_n=\sum_{i = 1}^{n}f(x_i)\Delta x$. Here, $L_3=f(x_1)\Delta x+f(x_2)\Delta x+f(x_3)\Delta x$. Substituting $\Delta x = 2$, $f(x_1)=5$, $f(x_2)=1$, $f(x_3)=5$ we get $L_3=2\times5+2\times1+2\times5=10 + 2+10=22$.

Answer:

22