use the midpoint rule with the given value of n to approximate the integral. round the answer to four…

Explanation:

Step1: Calculate the width of sub - intervals

The formula for the width of sub - intervals $\Delta x=\frac{b - a}{n}$, where $a = 1$, $b = 5$, and $n = 4$. So, $\Delta x=\frac{5 - 1}{4}=1$.

Step2: Determine the mid - points of sub - intervals

The sub - intervals are $[1,2]$, $[2,3]$, $[3,4]$, $[4,5]$. The mid - points $x_1 = 1.5$, $x_2=2.5$, $x_3 = 3.5$, $x_4 = 4.5$.

Step3: Evaluate the function at mid - points

Let $f(x)=5x^{5}e^{-x}$. Then $f(x_1)=5\times(1.5)^{5}e^{-1.5}$, $f(x_2)=5\times(2.5)^{5}e^{-2.5}$, $f(x_3)=5\times(3.5)^{5}e^{-3.5}$, $f(x_4)=5\times(4.5)^{5}e^{-4.5}$. $f(1.5)=5\times7.59375\times0.22313 = 5\times1.6947=8.4735$. $f(2.5)=5\times97.65625\times0.08208 = 5\times8.0147=40.0735$. $f(3.5)=5\times525.21875\times0.03020 = 5\times15.8615=79.3075$. $f(4.5)=5\times1845.28125\times0.01111 = 5\times20.4011=102.0055$.

Step4: Apply the Midpoint Rule

The Midpoint Rule is $M_n=\Delta x\sum_{i = 1}^{n}f(x_i)$. Here, $M_4=\Delta x(f(x_1)+f(x_2)+f(x_3)+f(x_4))$. $M_4=1\times(8.4735 + 40.0735+79.3075 + 102.0055)=229.8600$.

Answer:

$229.8600$