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.