current attempt in progress estimate ∫₀¹² 1/(x + 5) dx using a left - hand sum with n = 3. round your answer…

Explanation:

Step1: Calculate the width of sub - intervals

The interval is $[a,b]=[0,12]$ and $n = 3$. The width of each sub - interval $\Delta x=\frac{b - a}{n}=\frac{12-0}{3}=4$.

Step2: Determine the left - hand endpoints

The left - hand endpoints of the sub - intervals $[0,4]$, $[4,8]$, $[8,12]$ are $x_0 = 0$, $x_1=4$, $x_2 = 8$.

Step3: Calculate the left - hand sum

The left - hand sum $L_3=\sum_{i = 0}^{2}f(x_i)\Delta x$, where $f(x)=\frac{1}{x + 5}$. $f(x_0)=\frac{1}{0 + 5}=\frac{1}{5}$, $f(x_1)=\frac{1}{4+5}=\frac{1}{9}$, $f(x_2)=\frac{1}{8 + 5}=\frac{1}{13}$. $L_3=\Delta x\left(f(x_0)+f(x_1)+f(x_2)\right)=4\left(\frac{1}{5}+\frac{1}{9}+\frac{1}{13}\right)$. First, find a common denominator: The common denominator of 5, 9 and 13 is $5\times9\times13 = 585$. $\frac{1}{5}+\frac{1}{9}+\frac{1}{13}=\frac{9\times13+5\times13 + 5\times9}{585}=\frac{117+65 + 45}{585}=\frac{227}{585}$. Then $L_3=4\times\frac{227}{585}=\frac{908}{585}\approx1.552$.

Answer:

$1.552$