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Here’s a step-by-step explanation of the **Simpson's Rule** in the image:

### Step 1: Problem Setup
- You are asked to approximate the integral \( \int_1^2 \sqrt{x^3 - 1} \, dx \) using **Simpson’s Rule** with \( n = 10 \) subdivisions.
  
### Step 2: Formula for Simpson's Rule
- The Simpson's Rule formula is:
  \[
  \int_a^b f(x) dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \dots + 4f(x_{n-1}) + f(x_n) \right]
  \]
  where:
  - \( h = \frac{b - a}{n} = \frac{2 - 1}{10} = 0.1 \).
  - \( x_0 = a = 1 \), \( x_n = b = 2 \), and \( x_1, x_2, \dots, x_{n-1} \) are intermediate points.

### Step 3: Calculating Function Values
- The function being integrated is \( f(x) = \sqrt{x^3 - 1} \).
- The function is evaluated at 11 points: \( x_0 = 1, x_1 = 1.1, \dots, x_{10} = 2 \).
- Function values:
  \[
  f(1) = 0, \quad f(1.1) = 0.575326, \quad f(1.2) = 0.853229, \dots, f(2) = 2.64575
  \]

### Step 4: Applying Simpson's Rule
- Using the function values, apply Simpson’s Rule:
  \[
  \int_1^2 \sqrt{x^3 - 1} \, dx \approx \frac{0.1}{3} \left[ f(1) + 4f(1.1) + 2f(1.2) + 4f(1.3) + 2f(1.4) + \dots + 4f(1.9) + f(2) \right]
  \]
  Substituting the function values:
  \[
  \frac{0.1}{3} \left[ 0 + 4(0.575326) + 2(0.853229) + 4(1.09407) + \dots + 4(2.42054) + 2.

Explain the step-by-step solution using Simpson's Rule to approximate the integral of √(x^3 - 1) from 1 to 2, with n = 10 subdivisions.

This problem demonstrates how to apply Simpson's Rule to approximate the integral ∫√(x^3 - 1) dx from 1 to 2, using 10 subdivisions. The solution includes key concepts in numerical integration and a step-by-step application of the Simpson's Rule formula.

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