The correct answer is 63.

Based on the provided image, we are asked to find the value of the integral:
${��}_2^4{x}^{3}dx$
using a 2-equal-segment trapezoidal rule.

Let's break down the solution step-by-step:

Step 1: Identify the parameters from the integral and the method
The integration interval is $[a,b]=[2,4]$, so the lower limit is $a=2$ and the upper limit is $b=4$.
The number of segments is $n=2$.
The function to be integrated is:
$f\left(x\right)=x^3$

Step 2: Calculate the segment width (step size, $h$)
The formula for the step size $h$ is:
$h=\frac{b-a}{n}$
Substituting the given values:
$h=\frac{4-2}{2}=\frac{2}{2}=1$

Step 3: Determine the grid points
The grid points $x_i$ are defined as:
$x_i=a+i·h$
For $i=0,1,2$:
$x_0=2$
$x_1=2+1=3$
$x_2=2+2\left(1\right)=4$

Step 4: Evaluate the function at each grid point
Evaluate $f\left(x\right)=x^3$ at the grid points:
$f\left(x_0\right)=f\left(2\right)=2^3=8$
$f\left(x_1\right)=f\left(3\right)=3^3=27$
$f\left(x_2\right)=f\left(4\right)=4^3=64$

Step 5: Apply the multiple-segment Trapezoidal Rule formula
The multiple-segment trapezoidal rule formula for $n=2$ is given by:
$I\approx \frac{h}{2}\left[f\right(x_0)+2f(x_1)+f(x_2\left)\right]$
Substitute the values into the equation:
$I\approx \frac{1}{2}[8+2(27)+64]$
Simplify the expression inside the brackets:
$I\approx \frac{1}{2}[8+54+64]$
$I\approx \frac{1}{2}\left[126\right]$
$I\approx 63$

Thus, evaluating the integral using the 2-equal-segment trapezoidal rule yields a value of 63.