Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)
2

0
ex
9 + x2
 dx,    n = 10
(a) the Trapezoidal Rule
 
Incorrect: Your answer is incorrect.


(b) the Midpoint Rule
 
Incorrect: Your answer is incorrect.


(c) Simpson's Rule

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.

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) ∫ 0 to 2 of (e^x / (9 + x^2)) dx, n = 10.

Learn how to approximate the integral ∫ 0 to 2 of (e^x / (9 + x^2)) dx using the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule with n = 10 subintervals. This tutorial covers key numerical methods and their applications in calculus, providing step-by-step solutions and accuracy comparisons.

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