use the midpoint rule to approximate the following integral to four decimal places using four…

Explanation:

Step1: Calculate the width of sub - intervals

The formula for the width $\Delta x$ of sub - intervals is $\Delta x=\frac{b - a}{n}$, where $a = 3$, $b = 7$, and $n=4$. So, $\Delta x=\frac{7 - 3}{4}=1$.

Step2: Determine the mid - points of sub - intervals

The sub - intervals are $[3,4]$, $[4,5]$, $[5,6]$, $[6,7]$. The mid - points $x_i$ are $x_1 = 3.5$, $x_2=4.5$, $x_3 = 5.5$, $x_4=6.5$.

Step3: Evaluate the function at mid - points

Let $f(x)=3x^{3}e^{-x}$. Then $f(3.5)=3\times(3.5)^{3}\times e^{-3.5}\approx3\times42.875\times0.0302 = 3.9077$, $f(4.5)=3\times(4.5)^{3}\times e^{-4.5}\approx3\times91.125\times0.0111=3.0129$, $f(5.5)=3\times(5.5)^{3}\times e^{-5.5}\approx3\times166.375\times0.0041 = 2.0564$, $f(6.5)=3\times(6.5)^{3}\times e^{-6.5}\approx3\times274.625\times0.0015=1.2358$.

Step4: Apply the Midpoint Rule

The Midpoint Rule is $M_n=\Delta x\sum_{i = 1}^{n}f(x_i)$. So $M_4=1\times(3.9077 + 3.0129+2.0564 + 1.2358)=10.2128$.

Answer:

$10.2128$