To approximate the integral using Simpson's Rule with $n = 50$ and a lower limit of $a = 0.0001$, we need to use the formula for Simpson's Rule, which is:

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

Where:

• $a = 0.0001$ is the lower limit of the integral
• $b = \frac{\pi}{2}$ is the upper limit of the integral
• $n = 50$ is the number of subintervals
• $f(x) = \frac{4 \sin(x)}{x}$ is the integrand

We can break down the calculation steps:

1. Calculate the values of $x_i$ for each step, where $x_i = a + i \cdot \frac{b - a}{n}$, for $i = 0, 1, 2, \dots, n$.
2. Evaluate the function $f(x)$ at these points.
3. Apply Simpson’s Rule using the formula above.

I can calculate this for both $n = 50$ and $n = 100$ and give you the approximations. Let me run the computations. numpy as np

Define the function to integrate

def f(x): return 4 * np.sin(x) / x if x != 0 else 4 # Handling x=0

Define Simpson's Rule Approximation

def simpsons_rule(a, b, n): # Step size h = (b - a) / n

# x values
x = np.linspace(a, b, n+1)

# Function values
fx = np.array([f(xi) for xi in x])

# Apply Simpson's rule
integral = (h / 3) * (fx[0] + fx[-1] + 4 * np.sum(fx[1:-1:2]) + 2 * np.sum(fx[2:-2:2]))
return integral

Given parameters

a = 0.0001 # Lower limit b = np.pi / 2 # Upper limit

Calculating the approximations

n_50 = 50 n_100 = 100

simpson_50 = simpsons_rule(a, b, n_50) simpson_100 = simpsons_rule(a, b, n_100)

simpson_50, simpson_100The Simpson's Rule approximations for the integral are as follows:

• With $n = 50$, the approximation is approximately 5.4826.
• With $n = 100$, the approximation is approximately 5.4826 (it converges very closely to the value with more subintervals).

Would you like to explore more about Simpson's Rule, or do you have any questions about these results?

Here are a few related questions you might find interesting:

1. How does increasing $n$ further affect the accuracy of the approximation?
2. What is the error in Simpson's Rule, and how can it be estimated?
3. How would you apply Simpson's Rule for an integral with a different function?
4. Can Simpson's Rule be applied to improper integrals, and if so, how?
5. How does Simpson's Rule compare to the Trapezoidal Rule in terms of accuracy?

Tip: To achieve better accuracy with Simpson's Rule, ensure that your subintervals are evenly spaced, and you use an $n$ value that is a multiple of 2.