To approximate the integral $\int_3^9 \frac{8}{(s-2)^2} \, ds$ using Simpson's rule with an error less than $10^{-8}$, we calculate the smallest number of subintervals $n$ required based on Simpson's error estimate formula.

Error estimate formula for Simpson's rule:

$E_S = \frac{(b-a)^5}{180 n^4} \max |f^{(4)}(x)|,$ where:

• $[a, b]$ is the interval of integration ($a = 3, b = 9$),
• $n$ is the number of subintervals (must be even),
• $f(x)$ is the integrand ($f(x) = \frac{8}{(x-2)^2}$),
• $f^{(4)}(x)$ is the fourth derivative of $f(x)$.

Step 1: Find the fourth derivative $f^{(4)}(x)$:

The function is: $f(x) = \frac{8}{(x-2)^2}.$

First derivative: $f'(x) = -\frac{16}{(x-2)^3}.$

Second derivative: $f''(x) = \frac{48}{(x-2)^4}.$

Third derivative: $f^{(3)}(x) = -\frac{192}{(x-2)^5}.$

Fourth derivative: $f^{(4)}(x) = \frac{960}{(x-2)^6}.$

Step 2: Determine the maximum of $|f^{(4)}(x)|$ on $[3, 9]$:

On the interval $[3, 9]$, $f^{(4)}(x)$ is largest when $x$ is smallest (as $(x-2)^6$ is smallest). At $x = 3$: $f^{(4)}(3) = \frac{960}{(3-2)^6} = 960.$

Thus, $\max |f^{(4)}(x)| = 960$.

Step 3: Apply the error formula and solve for $n$:

The error formula becomes: $E_S \leq \frac{(9-3)^5}{180 n^4} \cdot 960.$

Simplify: $E_S \leq \frac{6^5 \cdot 960}{180 n^4}.$

Calculate $6^5 = 7776$: $E_S \leq \frac{7776 \cdot 960}{180 n^4}.$

Simplify the constant: $E_S \leq \frac{7464960}{180 n^4} = \frac{41472}{n^4}.$

Set $E_S < 10^{-8}$: $\frac{41472}{n^4} < 10^{-8}.$

Multiply through by $n^4$: $41472 < 10^{-8} n^4.$

Divide through by $10^{-8}$: $n^4 > \frac{41472}{10^{-8}} = 4.1472 \times 10^{12}.$

Take the fourth root: $n > (4.1472 \times 10^{12})^{1/4}.$

Step 4: Approximate $n$:

$(4.1472 \times 10^{12})^{1/4} \approx 3165.5.$

Since $n$ must be even, the smallest $n$ is: $n = 3166.$

Final Answer:

The smallest number of subintervals $n$ is $\boxed{3166}$.

Would you like a detailed explanation of any step, or do you have any further questions?

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Questions to Explore Further:

1. What is the reasoning behind choosing an even number of subintervals in Simpson's Rule?
2. How does the fourth derivative affect the error in Simpson's Rule?
3. What would the error estimate be if we used a smaller $n$?
4. Can you verify the result by numerically integrating $\frac{8}{(s-2)^2}$ with $n = 3166$?
5. How does Simpson's Rule compare to other numerical methods like the trapezoidal rule for this integral?

Tip:

When approximating integrals with Simpson's Rule, always ensure the integrand is sufficiently smooth (continuous derivatives) over the interval for the method to be effective.