find the riemann sum s4 for the following information. round your answer to the nearest hundredth. f(x)=49…

Explanation:

Step1: Calculate the width of sub - intervals

The formula for the width $\Delta x$ of sub - intervals on $[a,b]$ with $n$ sub - intervals is $\Delta x=\frac{b - a}{n}$. Here, $a=-7$, $b = 1$, and $n = 4$. So, $\Delta x=\frac{1-(-7)}{4}=\frac{8}{4}=2$.

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

For $i = 1$: $f(c_1)=49-( - 6.5)^2=49 - 42.25 = 6.75$. For $i = 2$: $f(c_2)=49-( - 4.5)^2=49 - 20.25 = 28.75$. For $i = 3$: $f(c_3)=49-( - 2.5)^2=49 - 6.25 = 42.75$. For $i = 4$: $f(c_4)=49-( - 0.5)^2=49 - 0.25 = 48.75$.

Step3: Calculate the Riemann sum $S_4$

The Riemann sum $S_n=\sum_{i = 1}^{n}f(c_i)\Delta x$. Here, $n = 4$ and $\Delta x=2$. $S_4=f(c_1)\Delta x+f(c_2)\Delta x+f(c_3)\Delta x+f(c_4)\Delta x=\Delta x\left(f(c_1)+f(c_2)+f(c_3)+f(c_4)\right)$. $S_4=2\times(6.75 + 28.75+42.75+48.75)$. $S_4=2\times127$. $S_4 = 254.00$.

Answer:

$254.00$