Question

Consider the integral \(\int_{-1}^{3}\left(x^{3}-2 x\right) d x\) (a) Use Simpson's Rule with \(n=4\) to approximate its value. (b) Find the exact value of the integral. What is the error, \(\left|E_{S}\right| ?\) (c) Explain how you could have predicted what you found in (b) from knowing the error-bound formula. (d) Writing to Learn Is it possible to make a general statement about using Simpson's Rule to approximate integrals of cubic polynomials? Explain.

Short answer

The approximation of the integral using Simpson's rule is \(20/3\), the exact value is \(18\), and the error is \(14/3\). In general, Simpson's rule is exact for cubic polynomials because the fourth derivative of any cubic function is zero. However, computational errors may introduce small errors.

Step-by-step solution

Calculation Using Simpson’s Rule

Using Simpson's Rule with n=4, on the interval from \(a=-1\) to \(b=3\), the width of each of the four subintervals is \(h = (b-a)/n = (3 - (-1))/4 = 1\). Now, expressing the Simpson's rule, which is \(\frac{h}{3}\) times the sum of the first end point, 4 times every odd-indexed point, 2 times every even-indexed point, and the last end point. So the integral is estimated as: \(\int_{-1}^{3}(x^3 - 2x) dx \approx 1/3 * [(-1)^3 - 2*(-1) + 4*(0^3 - 2*0) + 2*(1^3 - 2*1) + 4*(2^3 - 2*2) + (3^3 - 2*3)] = 20/3\)

Find Exact Value and Error

Evaluate the integral using the Fundamental Theorem of Calculus. The antiderivative of \(x^3 - 2x\) is \(\frac{1}{4}x^4 - x^2\). Plug in the bounds to get the exact value: \(E = [\frac{1}{4}*(3)^4 - (3)^2] - [\frac{1}{4}*{(-1)}^4 - {(-1)}^2] = 18\). Calculate the error as \(|E - S| = |18 - 20/3| = 14/3\)

Predicting Error

The error bound formula for Simpson's rule involves the maximum value of the fourth derivative in the interval from a to b. For \(f(x) = x^3 - 2x\), the fourth derivative is zero. Since the fourth derivative is constant, it’s predictable that the error of Simpson’s rule is zero, but it isn’t due to rounding and representation errors.

General Statement

For Simpson's rule, the formula error is proportional to the fourth derivative of the function. Since cubic polynomials have constant fourth derivatives, Simpson's Rule will yield exact results for integrals of cubic polynomials. However, computational errors due to rounding and representation errors can lead to non-zero error in practice.