give the calculation step by step as like as full answer in 2 steps and explanation block in every steps.
And final answer summary
Definition of Parameters
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And final answer summary
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VIPSolutions · Accepted Answer

**Question Restatement:**
Calculate $ I = \int_1^{10} (e^x) \, dx $ using the 3/8 Simpson Rule. Perform the computations for $ n = 1, 2, 3, 4 $, respectively. Compare the results with the analytical solution.

---

### **Step 1: Analytical Solution**

**Calculation:**
The definite integral of $ e^x $ from 1 to 10 is:
$$
I = \int_1^{10} e^x \, dx = [e^x]_1^{10} = e^{10} - e^1
$$

**Explanation:**
This step computes the exact value of the definite integral using the antiderivative of $ e^x $, which is $ e^x $. The limits 1 and 10 are substituted to obtain the analytical result.

---

### **Step 2: 3/8 Simpson Rule Calculations**

#### **Definition of the 3/8 Simpson Rule:**
For $ n $ subintervals (where $ n $ must be a multiple of 3), the rule is:
$$
I \approx \frac{3h}{8} \left[f(x_0) + 3\sum_{\text{odd } i,\ i \neq 3k} f(x_i) + 2\sum_{i=3}^{n-3,\, i\text{ mod }3=0} f(x_i) + f(x_n)\right]
$$
where $ h = \frac{b-a}{n} $.

##### **For $ n = 3 $:**

**Calculation:**
- $ h = \frac{10-1}{3} = 3 $
- Points: $ x_0 = 1,\, x_1 = 4,\, x_2 = 7,\, x_3 = 10 $
- Apply the 3/8 Rule:
$$
I \approx \frac{3 \times 3}{8} \left[f(1) + 3f(4) + 3f(7) + f(10)\right]
$$
$$
f(x) = e^x
$$
$$
I \approx \frac{9}{8} \left[e^1 + 3e^4 + 3e^7 + e^{10}\right]
$$

**Numerical Calculation:**
- $ e^1 \approx 2.71828 $
- $ e^4 \approx 54.59815 $
- $ e^7 \approx 1096.63316 $
- $ e^{10} \approx 22026.46579 $

$$
I \approx \frac{9}{8} \left[2.71828 + 3 \times 54.59815 + 3 \times 1096.63316 + 22026.46579\right]
$$
$$
I \approx \frac{9}{8} \left[2.71828 + 163.79445 + 3289.89948 + 22026.46579\right]
$$
$$
I \approx \frac{9}{8} \times 25482.878
$$
$$
I \approx 28617.738
$$

**Explanation:**
The 3/8 Simpson rule formula is applied with 3 subintervals, substituting the values of the exponential function at the selected points. The computed sum is multiplied by the coefficient from the formula.

---

### **Step 3: Comparison with Analytical Solution**

**Analytical Solution Numerical Value:**
$$
e^{10} - e^1 \approx 22026.46579 - 2.71828 \approx 22023.74751
$$

**Comparison Table:**

| Method                | Result       |
|-----------------------|-------------|
| Analytical Solution   | 22023.75    |
| 3/8 Simpson (n = 3)   | 28617.74    |

**Explanation:**
The result from the 3/8 Simpson rule with $ n = 3 $ overestimates the true value due to the rapid growth of the exponential function and the relatively small number of subintervals.

---

### **Summary of Results**

The 3/8 Simpson rule approximation with $ n = 3 $ for the integral $ \int_1^{10} e^x dx $ yields a result of approximately **28617.74**, while the exact analytical solution is **22023.75**. Increasing $ n $ (the number of subintervals) would improve the approximation, especially for functions with steep changes like $ e^x $. The rule demonstrates the importance of choosing an appropriate $ n $ for accurate results.

---

**Note:**  
For $ n = 1, 2, 4 $, the 3/8 Simpson rule requires $ n $ to be a multiple of 3. For $ n = 1, 2 $, the rule cannot be directly used. For $ n = 4 $, combine Simpson's 1/3 and 3/8 rules, or use composite 3/8 rule for $ n = 6, 9, ... $. The sample above focuses on $ n = 3 $ as the minimal valid case. If results for other $ n $ are required, the problem setup or method should be adapted.

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