Let's solve the problem step by step for both the Trapezoidal Rule and Simpson's Rule. --- ## **Given Integral** \[ \int_{0}^{8} \frac{1}{1+x} \, dx \] Let \( f(x) = \frac{1}{1+x} \). - \( a = 0 \), \( b = 8 \) - We want the error \( |E| \leq 0.00001 \). --- ## **Step 1: Compute Necessary Derivatives** ### For the **Trapezoidal Rule** (needs \( |f''(x)| \)): \[ f(x) = (1+x)^{-1} \] \[ f'(x) = - (1+x)^{-2} \] \[ f''(x) = 2(1+x)^{-3} \] - The error bound uses \( \max |f''(x)| \) on \( [0, 8] \). - \( f''(x) \) is **maximized at \( x = 0 \)** (since denominator increases as \( x \) increases). So, \[ |f''(0)| = 2 \] \[ |f''(x)| = \frac{2}{(1+x)^3} \] So, \( \max |f''(x)| = 2 \) on \( [0, 8] \). --- ### For the **Simpson's Rule** (needs \( |f^{(4)}(x)| \)): \[ f'''(x) = -6(1+x)^{-4} \] \[ f^{(4)}(x) = 24(1+x)^{-5} \] - Maximum at \( x = 0 \): \( |f^{(4)}(0)| = 24 \) - \( \max |f^{(4)}(x)| = 24 \) on \( [0, 8] \) --- ## **Step 2: Error Formulas** ### **Trapezoidal Rule:** \[ |E| \leq \frac{(b-a)^3}{12n^2} \max|f''(x)| \] Plug values in: \[ |E| \leq \frac{(8-0)^3}{12n^2} \cdot 2 = \frac{512}{12n^2} \cdot 2 = \frac{1024}{12n^2} = \frac{256}{3n^2} \] We want \( |E| \leq 0.00001 \): \[ \frac{256}{3n^2} \leq 0.00001 \] \[ 256 \leq 0.00001 \times 3n^2 \] \[ 256 \leq 0.00003 n^2 \] \[ n^2 \geq \frac{256}{0.00003} \] \[ n^2 \geq 8,533,333.33 \] \[ n \geq \sqrt{8,533,333.33} \approx 2,922 \] ### **Minimum integer \( n \) for Trapezoidal Rule:** \[ \boxed{n = 2,922} \] --- ### **Simpson's Rule:** \[ |E| \leq \frac{(b-a)^5}{180n^4} \max|f^{(4)}(x)| \] Plug values in: \[ |E| \leq \frac{(8-0)^5}{180n^4} \cdot 24 = \frac{32768}{180n^4} \cdot 24 = \frac{786432}{180n^4} = \frac{4369.0667}{n^4} \] We want \( |E| \leq 0.00001 \): \[ \frac{4369.0667}{n^4} \leq 0.00001 \] \[ 4369.0667 \leq 0.00001 n^4 \] \[ n^4 \geq \frac{4369.0667}{0.00001} \] \[ n^4 \geq 436,906,670 \] \[ n \geq (436,906,670)^{1/4} \] \[ n \geq 144.5 \] ### **Minimum integer \( n \) for Simpson's Rule:** *Simpson's Rule requires \( n \) to be **even**:* \[ \boxed{n = 146} \] (where 146 is the next even integer after 144.5) --- ## **Final Answers** ### **Minimum \( n \) such that error \( \leq 0.00001 \):** - **Trapezoidal Rule:**                                                                                   \(\boxed{2,922}\) - **Simpson's Rule (even \( n \)):**                                                             \(\boxed{146}\) --- If you need the approximated values of the integrals or more explanation, let me know!