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.) $$\int_{1}^{2} \frac{1}{x} d x$$

Short answer

The approximate value of the integral calculated using Simpson's rule and the exact value of the integral will be determined in the solution steps. The final step involves comparing those two values.

Step-by-step solution

Simpson's Rule Approximation

Simpson's rule is a numerical method for approximating the definite integral of a function. It's given by the formula: \[ \int_{a}^{b} f(x) dx \approx \frac{h}{3} [ f(a) + 4f(x1) + 2f(x2) + 4f(x3) + \ldots + 4f(x_{n-1}) + f(b)] \] Where \(h = (b-a)/n\), and \(x_{i} = a + ih\) for \(i = 0,1,2,...n\). Here, \(a=1\), \(b=2\), \(n=4\), so \(h = (2-1)/4 = 0.25\). The function to be integrated is \(f(x) = \frac{1}{x}\), so calculate \(f(a)\), \(f(x1)\), \(f(x2)\), \(f(x3)\), \(f(x4)\), and \(f(b)\). Then substitute these values into the Simpson's rule formula to compute the integral approximation.

Exact Integral Calculation

To calculate the exact value, you need to determine the antiderivative of the function \(f(x) = \frac{1}{x}\), which is \(F(x) = ln|x|\). The definite integral from a to b of f(x) dx is given by \(F(b) - F(a)\). In this case, you need to evaluate \(F(2) - F(1)\), which will give the exact value of the integral.

Comparing the Values

Finally, compare the values obtained from Simpson's rule and the exact integral calculation. Note that due to the numerical approximation method, there might be a slight difference in the values. But under ideal conditions and for well-behaved functions, the values should be very close. You can also compare this result with the previously computed result using the Trapezoidal rule as an additional verification.