1. **Problem 1:** Approximate the integral $$\int_0^8 \sqrt{x} \, dx$$ using Simpson's Rule with $n=4$ subintervals. 2. **Formula:** Simpson's Rule approximation is given by $$ S_n = \frac{\Delta x}{3} \left[f(x_0) + 4 \sum_{i=1,3,5,...}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,...}^{n-2} f(x_i) + f(x_n)\right] $$ where $\Delta x = \frac{b-a}{n}$. 3. **Step 1:** Calculate $\Delta x = \frac{8-0}{4} = 2$. 4. **Step 2:** Determine the $x_i$ values: $0, 2, 4, 6, 8$. 5. **Step 3:** Evaluate $f(x_i) = \sqrt{x_i}$: - $f(0) = 0$ - $f(2) = \sqrt{2}$ - $f(4) = 2$ - $f(6) = \sqrt{6}$ - $f(8) = 2\sqrt{2}$ 6. **Step 4:** Apply Simpson's Rule: $$ S_4 = \frac{2}{3} \left[0 + 4(\sqrt{2} + \sqrt{6}) + 2(2) + 2\sqrt{2}\right] $$ 7. **Step 5:** Simplify inside the brackets: $$ 4(\sqrt{2} + \sqrt{6}) = 4\sqrt{2} + 4\sqrt{6} $$ $$ 2(2) = 4 $$ So, $$ 0 + 4\sqrt{2} + 4\sqrt{6} + 4 + 2\sqrt{2} = 6\sqrt{2} + 4\sqrt{6} + 4 $$ 8. **Step 6:** Multiply by $\frac{2}{3}$: $$ S_4 = \frac{2}{3} (6\sqrt{2} + 4\sqrt{6} + 4) = 4\sqrt{2} + \frac{8}{3}\sqrt{6} + \frac{8}{3} $$ 9. **Step 7:** Approximate numerically: - $\sqrt{2} \approx 1.414$ - $\sqrt{6} \approx 2.449$ $$ S_4 \approx 4(1.414) + \frac{8}{3}(2.449) + \frac{8}{3} = 5.656 + 6.530 + 2.667 = 14.853 $$ --- 10. **Problem 2:** Approximate area under $y=f(x)$ from $x=0$ to $x=4$ with $n=4$ using Simpson's Rule. 11. **Given:** $$ x: 0,1,2,3,4 $$ $$ f(x): 2,7,12,10,5 $$ 12. **Step 1:** Calculate $\Delta x = \frac{4-0}{4} = 1$. 13. **Step 2:** Apply Simpson's Rule: $$ S_4 = \frac{1}{3} [f(0) + 4(f(1) + f(3)) + 2(f(2)) + f(4)] $$ 14. **Step 3:** Substitute values: $$ S_4 = \frac{1}{3} [2 + 4(7 + 10) + 2(12) + 5] = \frac{1}{3} [2 + 4(17) + 24 + 5] $$ 15. **Step 4:** Simplify: $$ 2 + 68 + 24 + 5 = 99 $$ 16. **Step 5:** Final approximation: $$ S_4 = \frac{99}{3} = 33 $$ --- 17. **Problem 3:** Approximate area under $y=f(x)$ from $x=-4$ to $x=8$ with $n=6$ using Simpson's Rule. 18. **Given:** $$ x: -4, -2, 0, 2, 4, 6, 8 $$ $$ f(x): 1, 3, 4, 4, 6, 9, 14 $$ 19. **Step 1:** Calculate $\Delta x = \frac{8 - (-4)}{6} = \frac{12}{6} = 2$. 20. **Step 2:** Apply Simpson's Rule: $$ S_6 = \frac{2}{3} [f(-4) + 4(f(-2) + f(2) + f(6)) + 2(f(0) + f(4)) + f(8)] $$ 21. **Step 3:** Substitute values: $$ S_6 = \frac{2}{3} [1 + 4(3 + 4 + 9) + 2(4 + 6) + 14] $$ 22. **Step 4:** Simplify inside brackets: $$ 4(16) = 64, \quad 2(10) = 20 $$ $$ 1 + 64 + 20 + 14 = 99 $$ 23. **Step 5:** Final approximation: $$ S_6 = \frac{2}{3} \times 99 = 66 $$ **Final answers:** - Q1: $\approx 14.853$ - Q2: $33$ - Q3: $66$