To find the approximation of the integral [tex]\(\int_5^7 \sqrt{x^2-3} \, dx\)[/tex] using Simpson's Rule with [tex]\(n = 4\)[/tex], follow these steps:
1. Define Functions and Parameters:
The function we will integrate is [tex]\(f(x) = \sqrt{x^2 - 3}\)[/tex]. The limits of integration are [tex]\(a = 5\)[/tex] and [tex]\(b = 7\)[/tex], with [tex]\(n = 4\)[/tex] intervals.
2. Calculate the Step Size [tex]\(h\)[/tex]:
[tex]\[ h = \frac{b - a}{n} = \frac{7 - 5}{4} = 0.5 \][/tex]
3. Create Points of Partition:
The [tex]\(x\)[/tex]-values where we will evaluate the function are:
[tex]\[ x_0 = 5, \quad x_1 = 5 + h = 5.5, \quad x_2 = 6, \quad x_3 = 6.5, \quad x_4 = 7 \][/tex]
4. Evaluate [tex]\(f(x)\)[/tex] at These Points:
Compute [tex]\(f(x_i)\)[/tex] for [tex]\(i = 0, 1, 2, 3, 4\)[/tex]:
[tex]\[
f(x_0) = \sqrt{5^2 - 3} = \sqrt{22} \approx 4.6904
\][/tex]
[tex]\[
f(x_1) = \sqrt{5.5^2 - 3} = \sqrt{27.25} \approx 5.2202
\][/tex]
[tex]\[
f(x_2) = \sqrt{6^2 - 3} = \sqrt{33} \approx 5.7446
\][/tex]
[tex]\[
f(x_3) = \sqrt{6.5^2 - 3} = \sqrt{39.25} \approx 6.2663
\][/tex]
[tex]\[
f(x_4) = \sqrt{7^2 - 3} = \sqrt{46} \approx 6.7823
\][/tex]
5. Apply Simpson's Rule:
Simpson's Rule states:
[tex]\[ S_n = \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4) \right] \][/tex]
6. Plug in Function Values:
[tex]\[
S_n = \frac{0.5}{3} \left[ 4.6904 + 4(5.2202) + 2(5.7446) + 4(6.2663) + 6.7823 \right]
\][/tex]
[tex]\[
S_n = \frac{0.5}{3} \left[ 4.6904 + 20.8808 + 11.4892 + 25.0652 + 6.7823 \right]
\][/tex]
[tex]\[
S_n = \frac{0.5}{3} \left[ 68.908 \right]
\][/tex]
7. Calculate Integral Approximation:
[tex]\[
S_n = \frac{0.5 \times 68.908}{3} = \frac{34.454}{3} \approx 11.484
\][/tex]
Thus, the value of [tex]\(S_n\)[/tex] is [tex]\(\boxed{11.484}\)[/tex].
