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solve with the full solution using rectangular, trapezoidal, and Simpson’s

The problem involves evaluating the definite integral 

\[
\int_{1}^{3} e^{\frac{1}{x}} \, dx
\]

using three numerical methods: the rectangular method, the trapezoidal method, and Simpson’s rule. The number of subintervals, \( n \), is given as 10. Here's the step-by-step solution for each method:

### 1. Rectangular Method (Midpoint Rule)

The rectangular (or midpoint) rule approximates the area under the curve by using rectangles. 

1. **Interval Width:**

   The interval width \( h \) is given by:

   \[
   h = \frac{b - a}{n} = \frac{3 - 1}{10} = 0.2
   \]

2. **Midpoints Calculation:**

   The midpoints \( x_i \) are calculated as:

   \[
   x_i = a + \left(i - 0.5\right)h, \quad i = 1, 2, \ldots, n
   \]

3. **Approximating the Integral:**

   The rectangular approximation \( R \) is given by:

   \[
   R \approx h \sum_{i=1}^{n} f(x_i)
   \]

### 2. Trapezoidal Rule

The trapezoidal rule approximates the area under the curve by using trapezoids.

1. **Formula:**

   The trapezoidal approximation \( T \) is:

   \[
   T \approx \frac{h}{2} \left[f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b)\right]
   \]

2. **Calculating Function Values:**

   Calculate the function values \( f(x) = e^{1/x} \) at each point, including the endpoints.

### 3. Simpson’s Rule

Simpson’s rule approximates the integral by using parabolic segments.

1. **Formula:**

   Since \( n \) is even, we can use Simpson’s formula:

   \[
   S \approx \frac{h}{3} \left[f(a) + 4\sum_{\text{odd } i=1}^{n-1} f(x_i) + 2\sum_{\text{even } i=2}^{n-2} f(x_i) + f(b)\right]
   \]

Now let's compute these values numerically.Here are the results for the integral \(\int_{1}^{3} e^{\frac{1}{x}} \, dx\) using different numerical methods with \( n = 10 \):

1. **Rectangular Method (Midpoint Rule):**  
   Approximation: \( R \approx 3.5176 \)

2. **Trapezoidal Rule:**  
   Approximation: \( T \approx 3.5302 \)

3. **Simpson’s Rule:**  
   Approximation: \( S \approx 3.5220 \)

These methods give slightly different approximations of the integral, with Simpson’s rule generally being the most accurate among the three.

Would you like further details or have any questions? Here are some related questions to expand your understanding:

1. How does increasing the number of subintervals \( n \) affect the accuracy of these methods?
2. What are the error formulas for each of these numerical methods?
3. How do we choose which numerical method to use in practice?
4. What are the advantages of Simpson's rule over the other methods?
5. Can these methods be applied to improper integrals?

**Tip:** Simpson’s rule generally provides more accurate results than the rectangular and trapezoidal methods, especially when the function is smooth and well-behaved over the interval.

Solve with the full solution using rectangular, trapezoidal, and Simpson’s methods for the integral ∫[1 to 3] e^(1/x) dx, with n=10.

Explore the detailed solution to the integral ∫[1 to 3] e^(1/x) dx using numerical methods such as the Rectangular (Midpoint) rule, Trapezoidal rule, and Simpson's rule with 10 subintervals. Learn step-by-step how to apply these methods in calculus.

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