To solve this problem using Simpson's Rule, we need to:

1. Estimate the required number of subintervals $n$ so that the error is within $0.001$.
2. Approximate the integral of $\sin(x)$ from 0 to $\pi$ using Simpson's Rule with the calculated $n$.

Step 1: Error Bound for Simpson's Rule

The error for Simpson’s Rule is given by the formula:

$|E_S| \leq \frac{M(b - a)^5}{180n^4}$

Where:

• $E_S$ is the error of the approximation.
• $M$ is an upper bound for the fourth derivative of the function on the interval.
• $a = 0$, $b = \pi$ are the limits of integration.
• $n$ is the number of subintervals, which must be even.
• $180$ is a constant from Simpson's Rule error formula.

Finding $M$ (the maximum of the fourth derivative):

• The fourth derivative of $\sin(x)$ is simply $\sin(x)$, since the sine function repeats its derivatives every four steps.
• The maximum value of $|\sin(x)|$ over the interval $[0, \pi]$ is 1, so $M = 1$.

Thus, the error bound becomes:

$|E_S| \leq \frac{\pi^5}{180n^4}$

We want the error to be less than $0.001$, so we set the inequality:

$\frac{\pi^5}{180n^4} \leq 0.001$

Step 2: Solving for $n$

Now, solve this inequality for $n$:

$n^4 \geq \frac{\pi^5}{0.001 \times 180}$

Let me compute the right-hand side of this inequality.From the calculation, $n^4 \geq 6.42$, meaning that $n$ should be at least 7. However, since Simpson's Rule requires $n$ to be even, the smallest possible value of $n$ is $8$.

Step 3: Approximation using Simpson's Rule

Now, let's approximate the integral of $\sin(x)$ from $0$ to $\pi$ using Simpson's Rule with $n = 8$.

Simpson's Rule formula is:

$\int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \dots + f(x_n) \right]$

Where:

• $h = \frac{b - a}{n}$
• $x_0 = a$, $x_n = b$
• $x_i = a + i \cdot h$ for $i = 0, 1, \dots, n$

Let me calculate the Simpson's Rule approximation for $\int_0^\pi \sin(x) \, dx$.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can still follow these steps to approximate the integral of $\sin(x)$ using Simpson's Rule:

1. Divide the interval $[0, \pi]$ into $n = 8$ subintervals.
2. Use the formula for Simpson’s Rule to compute:

$\int_0^\pi \sin(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \dots + f(x_n) \right]$

Where $h = \frac{\pi - 0}{8} = \frac{\pi}{8}$.

This will give you an accurate approximation for the integral. If you need further clarification or would like help with something else, feel free to ask!

Relative Questions:

1. How does the error formula change if we use the trapezoidal rule instead of Simpson's Rule?
2. How does Simpson’s Rule perform compared to other numerical integration techniques for smooth functions like sine?
3. Can Simpson’s Rule handle discontinuous functions, and how would the error behave?
4. What is the geometric interpretation of Simpson’s Rule?
5. How would increasing $n$ further impact the accuracy of Simpson's Rule?

Tip:

For functions with higher oscillations or non-smooth behavior, increasing $n$ can drastically improve Simpson’s Rule accuracy!