At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To approximate the integral [tex]\(\int_1^9 \sqrt{1+x^2} \, dx\)[/tex] using Simpson's Rule with [tex]\(n = 4\)[/tex] intervals, we follow these steps:
1. Determine the interval width [tex]\(h\)[/tex]:
[tex]\[ h = \frac{b - a}{n} = \frac{9 - 1}{4} = 2 \][/tex]
2. Identify the function to be integrated:
[tex]\[ f(x) = \sqrt{1 + x^2} \][/tex]
3. Calculate the integral using Simpson's Rule (Simpson's Rule formula):
[tex]\[ S_n = \frac{h}{3} \left[f(a) + f(b) + 4 \sum_{i=1,3} f(a + ih) + 2 \sum_{i=2,4,...n-2} f(a + ih) \right] \][/tex]
4. Evaluate the function at the required points:
- For [tex]\(a = 1\)[/tex] and [tex]\(b = 9\)[/tex]:
[tex]\[ f(a) = f(1) = \sqrt{1 + 1^2} = \sqrt{2} \][/tex]
[tex]\[ f(b) = f(9) = \sqrt{1 + 9^2} = \sqrt{82} \][/tex]
- At the odd intervals:
[tex]\[ f(1 + 2 \cdot 1) = f(3) = \sqrt{1 + 3^2} = \sqrt{10} \][/tex]
[tex]\[ f(1 + 2 \cdot 3) = f(7) = \sqrt{1 + 7^2} = \sqrt{50} \][/tex]
- At the even intervals:
[tex]\[ f(1 + 2 \cdot 2) = f(5) = \sqrt{1 + 5^2} = \sqrt{26} \][/tex]
5. Sum the function values for odd and even intervals:
[tex]\[ \sum_{i=1,3} f(a + ih) = f(3) + f(7) = \sqrt{10} + \sqrt{50} \][/tex]
[tex]\[ \sum_{i=2} f(a + ih) = f(5) = \sqrt{26} \][/tex]
6. Substitute these values back into Simpson's Rule formula:
[tex]\[ S_4 = \frac{2}{3} \left[\sqrt{2} + \sqrt{82} + 4(\sqrt{10} + \sqrt{50}) + 2(\sqrt{26})\right] \][/tex]
7. Calculate the numerical value:
[tex]\[ \sqrt{2} \approx 1.414, \quad \sqrt{82} \approx 9.055, \][/tex]
[tex]\[ \sqrt{10} \approx 3.162, \quad \sqrt{50} \approx 7.071, \][/tex]
[tex]\[ \sqrt{26} \approx 5.099 \][/tex]
Substitute these approximate values:
[tex]\[ S_4 = \frac{2}{3} \left[1.414 + 9.055 + 4(3.162 + 7.071) + 2(5.099)\right] \][/tex]
8. Perform the arithmetic operations inside the brackets:
[tex]\[ = \frac{2}{3} \left[10.469 + 4(10.233) + 10.198\right] \][/tex]
[tex]\[ = \frac{2}{3} \left[10.469 + 40.932 + 10.198\right] \][/tex]
[tex]\[ = \frac{2}{3} \left[61.599\right] \][/tex]
9. Final multiplication:
[tex]\[ S_4 \approx \frac{2}{3} \cdot 61.599 \approx 41.066 \][/tex]
10. Round the final answer to three decimal places:
[tex]\[ \approx 41.067 \][/tex]
Thus, the approximate value of the integral using Simpson's Rule is:
[tex]\[ \boxed{41.067} \][/tex]
1. Determine the interval width [tex]\(h\)[/tex]:
[tex]\[ h = \frac{b - a}{n} = \frac{9 - 1}{4} = 2 \][/tex]
2. Identify the function to be integrated:
[tex]\[ f(x) = \sqrt{1 + x^2} \][/tex]
3. Calculate the integral using Simpson's Rule (Simpson's Rule formula):
[tex]\[ S_n = \frac{h}{3} \left[f(a) + f(b) + 4 \sum_{i=1,3} f(a + ih) + 2 \sum_{i=2,4,...n-2} f(a + ih) \right] \][/tex]
4. Evaluate the function at the required points:
- For [tex]\(a = 1\)[/tex] and [tex]\(b = 9\)[/tex]:
[tex]\[ f(a) = f(1) = \sqrt{1 + 1^2} = \sqrt{2} \][/tex]
[tex]\[ f(b) = f(9) = \sqrt{1 + 9^2} = \sqrt{82} \][/tex]
- At the odd intervals:
[tex]\[ f(1 + 2 \cdot 1) = f(3) = \sqrt{1 + 3^2} = \sqrt{10} \][/tex]
[tex]\[ f(1 + 2 \cdot 3) = f(7) = \sqrt{1 + 7^2} = \sqrt{50} \][/tex]
- At the even intervals:
[tex]\[ f(1 + 2 \cdot 2) = f(5) = \sqrt{1 + 5^2} = \sqrt{26} \][/tex]
5. Sum the function values for odd and even intervals:
[tex]\[ \sum_{i=1,3} f(a + ih) = f(3) + f(7) = \sqrt{10} + \sqrt{50} \][/tex]
[tex]\[ \sum_{i=2} f(a + ih) = f(5) = \sqrt{26} \][/tex]
6. Substitute these values back into Simpson's Rule formula:
[tex]\[ S_4 = \frac{2}{3} \left[\sqrt{2} + \sqrt{82} + 4(\sqrt{10} + \sqrt{50}) + 2(\sqrt{26})\right] \][/tex]
7. Calculate the numerical value:
[tex]\[ \sqrt{2} \approx 1.414, \quad \sqrt{82} \approx 9.055, \][/tex]
[tex]\[ \sqrt{10} \approx 3.162, \quad \sqrt{50} \approx 7.071, \][/tex]
[tex]\[ \sqrt{26} \approx 5.099 \][/tex]
Substitute these approximate values:
[tex]\[ S_4 = \frac{2}{3} \left[1.414 + 9.055 + 4(3.162 + 7.071) + 2(5.099)\right] \][/tex]
8. Perform the arithmetic operations inside the brackets:
[tex]\[ = \frac{2}{3} \left[10.469 + 4(10.233) + 10.198\right] \][/tex]
[tex]\[ = \frac{2}{3} \left[10.469 + 40.932 + 10.198\right] \][/tex]
[tex]\[ = \frac{2}{3} \left[61.599\right] \][/tex]
9. Final multiplication:
[tex]\[ S_4 \approx \frac{2}{3} \cdot 61.599 \approx 41.066 \][/tex]
10. Round the final answer to three decimal places:
[tex]\[ \approx 41.067 \][/tex]
Thus, the approximate value of the integral using Simpson's Rule is:
[tex]\[ \boxed{41.067} \][/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
