find the riemann sum s4 for the following information. round your answer to the nearest hundredth. f(x) =…

Explanation:

Step1: Calculate the width of sub - intervals

The formula for the width $\Delta x$ of sub - intervals is $\Delta x=\frac{b - a}{n}$. Here, $a=-8$, $b = - 4$, and $n = 4$. So, $\Delta x=\frac{-4-(-8)}{4}=\frac{-4 + 8}{4}=1$.

Step2: Calculate $f(c_i)$ for each $i$

For $i = 1$, $c_1=-7.5$, $f(c_1)=\frac{1}{-7.5 + 9}=\frac{1}{1.5}=\frac{2}{3}$. For $i = 2$, $c_2=-6.5$, $f(c_2)=\frac{1}{-6.5 + 9}=\frac{1}{2.5}=\frac{2}{5}$. For $i = 3$, $c_3=-5.5$, $f(c_3)=\frac{1}{-5.5 + 9}=\frac{1}{3.5}=\frac{2}{7}$. For $i = 4$, $c_4=-4.5$, $f(c_4)=\frac{1}{-4.5 + 9}=\frac{1}{4.5}=\frac{2}{9}$.

Step3: Calculate the Riemann sum $S_4$

The Riemann sum $S_n=\sum_{i = 1}^{n}f(c_i)\Delta x$. Since $\Delta x = 1$ and $n = 4$, $S_4=f(c_1)\Delta x+f(c_2)\Delta x+f(c_3)\Delta x+f(c_4)\Delta x$. $S_4=\left(\frac{2}{3}+\frac{2}{5}+\frac{2}{7}+\frac{2}{9}\right)\times1$. First, find a common denominator, which is $315$. $\frac{2}{3}+\frac{2}{5}+\frac{2}{7}+\frac{2}{9}=\frac{2\times105}{3\times105}+\frac{2\times63}{5\times63}+\frac{2\times45}{7\times45}+\frac{2\times35}{9\times35}=\frac{210}{315}+\frac{126}{315}+\frac{90}{315}+\frac{70}{315}=\frac{210 + 126+90 + 70}{315}=\frac{496}{315}\approx1.58$.

Answer:

$1.58$