The question requires using the trapezoidal rule to integrate a function given in Equation 2 between the range $0.1$ to $2.0$ with a step size of $0.1$. Let’s break this down:

Trapezoidal Rule Formula:

The trapezoidal rule is given by: $A = \sum_{i=1}^{n-1} \frac{y_{i+1} + y_i}{2} (x_{i+1} - x_i)$ where:

• $y_i$ and $y_{i+1}$ are the function values at consecutive points $x_i$ and $x_{i+1}$.
• $x_i$ and $x_{i+1}$ are the $x$-values in your interval.

Steps to Solve:

1. Determine the function from Equation 2.
   You will use the function provided in Equation 2 (which is not visible here), and calculate its values at points $x = 0.1, 0.2, 0.3, \ldots, 2.0$.

2. Apply the trapezoidal rule:

   • Use the step size $h = 0.1$.
   • Compute the function values at each $x_i$.
   • Sum the results using the formula.

3. Compare with Analytical Solution:
   The exercise mentions the analytical solution is $10.0792$. You can use this value to verify the accuracy of your trapezoidal approximation.

4. Optional: Try Simpson’s 1/3 Rule
   If you also need to try Simpson's rule, you may have to increase the number of intervals to an even number, since Simpson's rule requires it.

Would you like help with the specific computations of the trapezoidal rule, or do you have Equation 2 on hand to proceed? I can walk you through step-by-step!

Related Questions:

1. How do you determine if the trapezoidal approximation is close enough to the exact solution?
2. What are the conditions under which Simpson’s 1/3 rule performs better than the trapezoidal rule?
3. How does the error in numerical integration decrease with smaller step sizes?
4. What is the difference between composite and basic trapezoidal rule?
5. How can we extend the trapezoidal rule for unevenly spaced points?

Tip: Simpson's 1/3 rule generally provides better accuracy than the trapezoidal rule for smooth functions, but requires an even number of sub-intervals.