Question

The table lists several measurements gathered in an experiment to approximate an unknown continuous function \(y=f(x)\). (a) Approximate the integral \(\int_{0}^{2} f(x) d x\) using the Trapezoidal Rule and Simpson's Rule. \begin{tabular}{|c|c|c|c|c|c|} \hline\(x\) & 0.00 & 0.25 & 0.50 & 0.75 & 1.00 \\ \hline\(y\) & 4.32 & 4.36 & 4.58 & 5.79 & 6.14 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|c|} \hline\(x\) & 1.25 & 1.50 & 1.75 & 2.00 \\ \hline\(y\) & 7.25 & 7.64 & 8.08 & 8.14 \\ \hline \end{tabular} (b) Use a graphing utility to find a model of the form \(y=a x^{3}+b x^{2}+c x+d\) for the data. Integrate the resulting polynomial over [0,2] and compare your result with your results in part (a).

Short answer

The approximation of the integral from 0 to 2 of \(f(x)\) using the Trapezoidal Rule and Simpson's Rule, as well as the integral of the best-fit cubic equation over the same interval, are found in steps 1, 2, and 4, respectively. The comparison of these results is done in step 5.

Step-by-step solution

Applying the Trapezoidal Rule

Apply the Trapezoidal Rule as follows: integrate from 0 to 2 of \(f(x)\) = (b-a)/2n * [f(x0) + 2f(x1) + 2f(x2) +...+ 2f(xn-1) + f(xn)]. Here \(a=0\), \(b=2\), and \(n=8\) since we divide the interval into 8 parts. Substitute these values and all given \(f(xi)\) values to get the approximation.

Applying Simpson's Rule

Apply Simpson's Rule which is given as: integrate from 0 to 2 of \(f(x)\) = (b-a)/3n * [f(x0) + 4f(x1) + 2f(x2) +...+ 4f(xn-1) + f(xn)]. Here \(a=0\), \(b=2\), and \(n=8\). Substitute these values as well as the given \(f(xi)\) values to get the approximation.

Finding the coefficients for the cubic equation

Using a graphing utility, find a best fit for the function of the form \(f(x) = ax^3 + bx^2 + cx + d\) based on the given data points. The graphing utility will output the coefficients a, b, c, and d.

Integrate the cubic equation

Integrate the resulting cubic equation from 0 to 2. Remember the integral of \(ax^3\) is \((a/4)x^4\), of \(bx^2\) is \((b/3)x^3\), of \(cx\) is \((c/2)x^2\), and of \(d\) is \(dx\). Substitute \(x = 2\) and \(x = 0\) into these equations to find the definite integral.

Comparing results

Compare the result from the integral in Step 4 with the results from the Trapezoidal and Simpson's Rules in Steps 1 and 2. Note the differences and similarities between them.