give a 4 - term left riemann sum approximation for the integral below. \n\\(\\int_{10}^{14}7\\sqrt{x…

Explanation:

Step1: Calculate the width of each sub - interval

The interval is $[a,b]=[10,14]$ and $n = 4$. The width $\Delta x=\frac{b - a}{n}=\frac{14 - 10}{4}=1$.

Step2: Determine the left - hand endpoints

The left - hand endpoints of the sub - intervals are $x_0=a = 10$, $x_1=a+\Delta x=11$, $x_2=a + 2\Delta x=12$, $x_3=a+3\Delta x=13$.

Step3: Evaluate the function at the left - hand endpoints

The function is $f(x)=7\sqrt{x - 2}$. $f(x_0)=7\sqrt{10 - 2}=7\sqrt{8}=14\sqrt{2}$; $f(x_1)=7\sqrt{11 - 2}=7\sqrt{9}=21$; $f(x_2)=7\sqrt{12 - 2}=7\sqrt{10}$; $f(x_3)=7\sqrt{13 - 2}=7\sqrt{11}$.

Step4: Calculate the left Riemann sum

The left Riemann sum $L_4=\sum_{i = 0}^{3}f(x_i)\Delta x$. Since $\Delta x = 1$, $L_4=f(x_0)\times1+f(x_1)\times1+f(x_2)\times1+f(x_3)\times1=14\sqrt{2}+21 + 7\sqrt{10}+7\sqrt{11}$.

Answer:

$14\sqrt{2}+21 + 7\sqrt{10}+7\sqrt{11}$