Question

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral. Let \(n=4\) and round your answers to four decimal places. Use a graphing utility to verify your result. $$ \int_{0}^{4} \frac{8 x}{x^{2}+4} d x $$

Short answer

The results will be the numeric approximations of the definite integral from applying the Trapezoidal Rule and Simpson's Rule, calculated up to four decimal places and then verified graphically.

Step-by-step solution

Identify the Function and Interval

The Function given is \( f(x) = \frac{8x}{x^2 + 4} \) and the interval of integration is \([0, 4]\)

Apply the Trapezoidal Rule

The Trapezoidal Rule approximation is given by \(T_n= \frac{b - a} {2n} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]\). Here, `a` is the lower limit and `b` is the upper limit of the integral. `n` is the number of divisions of the interval chosen (here, \(n=4\)). So, first, calculate \( x_i = a + i*\frac{b-a}{n} \) for \( i = 0, 1, 2, ..., 4\). Then substitute \( x_i \) into \( f(x) \) to get \( f(x_i) \). Finally, substitute \( a, b, n, f(x_i) \) into the Trapezoidal Rule equation to evaluate the estimate of the integral.

Apply Simpson's Rule

The Simpson's Rule approximation is given by \( S_n = \frac{b - a} {3n} [f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n)] \). As defined before, use the calculated \( x_i \) and substitute into the function, then into the Simpson's Rule formula.

Round to four decimal places

Once both calculations for Trapezoidal Rule and Simpson's Rule are done, remember to round both of the results to four decimal places as asked in the problem.

Verify with a Graphing Utility

Finally, use a graphing method to plot the original function and find the areas under the curve with the Trapezoidal and Simpson's Rule methods. The numerical results should match with the graphical analysis.