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]
