Question

In Exercises 13-18, (a) use Simpson's Rule with n = 4 to approximate the value of the integral and (b) find the exact value of the integral to check your answer. (Note that these are the same integrals as Exercises 1-6, so you can also compare it with the Trapezoidal Rule approximation.) $$\int_{0}^{4} \sqrt{x} d x$$

Short answer

Simpson's rule estimate for the integral is \( \frac{8}{3} + \frac{2\sqrt{2}}{3} + \frac{4\sqrt{3}}{3} \), and the exact value of the integral is \( \frac{16}{3} \)

Step-by-step solution

Applying Simpson's Rule

First, divide the interval from 0 to 4 into 4 equally spaced intervals, resulting in h = (4-0)/4 = 1, and x_i = i*h for i from 0 to 4. The values are thus \(x_0 = 0\), \(x_1 = 1\), \(x_2 = 2\), \(x_3 = 3\), \(x_4 = 4\). Then, the Simpson's Rule formula, which is \(\frac{h}{3} (f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4))\), can be applied to estimate the integral.

Calculate Approximation

Calculate the function values \(f(x_i) = \sqrt{x_i}\) and substitute them into the Simpson's Rule formula to find the estimated integral value: \[\frac{1}{3} (\sqrt{0} + 4*\sqrt{1} + 2*\sqrt{2} + 4*\sqrt{3} + \sqrt{4})\] = \(\frac{1}{3} * (4 + 2\sqrt{2} + 4\sqrt{3} + 2) = \frac{8}{3} + \frac{2\sqrt{2}}{3} + \frac{4\sqrt{3}}{3}\]

Evaluate the Exact Integral

Integral of \( \sqrt{x} \) (= \( x^{1/2} \)) from 0 to 4 gives \(\frac{2}{3} x^{3/2}\) evaluated from 0 to 4. Substituting the limits of integration gives: \(\frac{2}{3} *(4^{3/2}) -\frac{2}{3} *(0^{3/2}) = \frac{16}{3}\)

Evaluate Approximation Error

To evaluate the error in the approximation, compare the Simpson's rule result from Step 2 and the exact value from Step 3. The exact value of the integral (16/3) is slightly different from the approximation, indicating a small error in approximation. This difference may be due to the fact that Simpson's rule is more accurate when the integrand is exactly a quadratic function, which is not the case here.