Let's compute the approximate value of the integral:

$\int_0^2 \frac{e^x}{9 + x^2} \, dx$

using the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule with $n = 10$.

Step-by-Step Process:

1. Define the function: $f(x) = \frac{e^x}{9 + x^2}$

2. Divide the interval $[0, 2]$ into $n = 10$ subintervals: $\Delta x = \frac{2 - 0}{10} = 0.2$

(a) Trapezoidal Rule

The Trapezoidal Rule is given by: $T_n = \frac{\Delta x}{2} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right]$

(b) Midpoint Rule

The Midpoint Rule is given by: $M_n = \Delta x \sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_i}{2}\right)$

(c) Simpson's Rule

Simpson's Rule is given by: $S_n = \frac{\Delta x}{3} \left[f(x_0) + 4 \sum_{i=1,3,5,\ldots}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,\ldots}^{n-2} f(x_i) + f(x_n)\right]$

Let's now compute these values.Here are the approximations for the integral $\int_0^2 \frac{e^x}{9 + x^2} \, dx$ using $n = 10$:

(a) Trapezoidal Rule: $T_n \approx 0.589164$

(b) Midpoint Rule: $M_n \approx 0.587753$

(c) Simpson's Rule: $S_n \approx 0.588225$

Would you like further details or clarifications?

Here are 5 questions to consider:

1. How does the accuracy of the Midpoint Rule compare to the Trapezoidal Rule?
2. Why is Simpson's Rule generally more accurate than the other two methods?
3. What happens to the accuracy of each rule as $n$ increases?
4. How can you derive the formulas for these numerical methods?
5. Are there scenarios where the Trapezoidal or Midpoint Rule might outperform Simpson's Rule?

Tip: For most problems, Simpson's Rule often gives the best results, but it requires an even number of intervals.