Question

Find an upper bound for the error in estimating \(\int_{0}^{3}\left(6 x^{2}-1\right) d x\) using Simpson's rule with \(n=10\) steps.

Short answer

The upper bound for the error is 0.

Step-by-step solution

- Define Simpson's Rule Error Bound

The error bound for Simpson's Rule is given by: \[ E_S = \frac{-(b-a)^5}{180n^4} \cdot M \] where \( M \) is the maximum value of the fourth derivative, \( n \) is the number of subintervals, and \( a \) and \( b \) are the limits of integration.

- Calculate the Fourth Derivative

The given function is \( f(x) = 6x^2 - 1 \). We need to find the fourth derivative, \( f^{(4)}(x) \). Calculating derivatives, we have: - First derivative: \( f'(x) = 12x \) - Second derivative: \( f''(x) = 12 \) - Third derivative: \( f^{(3)}(x) = 0 \) - Fourth derivative: \( f^{(4)}(x) = 0 \).Hence, \( M = 0 \) because the fourth derivative is constant and zero.

- Apply Error Bound Formula

Substitute the values into the error bound formula: \[ E_S = \frac{-(3-0)^5}{180 \times 10^4} \cdot 0 = 0 \] Since \( M = 0 \), the error bound becomes \( 0 \), indicating no error in estimation using Simpson's rule for this function.

Key concepts

Simpson's Rule

Simpson's Rule is a technique used to approximate the integral of a function. It is particularly effective when you want a more accurate estimate than basic trapezoidal or midpoint rules can provide.
It is based on the assumption that the function being integrated can be approximated by a series of parabolas over subintervals. Simpson's Rule averages the result of integrating the parabolic approximations.
For a function \( f(x) \) over the interval \([a, b]\) divided into \( n \) subintervals, Simpson's Rule is given by:

• Divide the interval into an even number of subintervals \( n \)
• Apply the formula: \[ S_n = rac{ riangle x}{3} (f(x_0) + 4 f(x_1) + 2 f(x_2) + \ \cdots + 4 f(x_{n-1}) + f(x_n)) \]

where \( \triangle x = \frac{b-a}{n} \).
This rule tends to be very accurate for polynomial functions up to the third degree or when the function is smooth over the interval.

Integral Estimation

Integral estimation is the process of finding approximate values for definite integrals. For functions that are difficult to integrate analytically or for which an antiderivative is not easily obtainable, numerical methods like Simpson's Rule are valuable.
These techniques are critical for problems across mathematics, physics, and engineering, where exact solutions are either impossible or impractical.
Simpson’s Rule stands out because it blends the simplicity of midpoint and trapezoidal approaches with the enhanced accuracy of considering curvature via parabolas. By doing so, it often requires fewer intervals to achieve a high level of accuracy than other methods.
When using Simpson's Rule, you transform the area under the curve into known geometric shapes (parabolas), and since parabolas fit the curves of many natural functions closely, the estimation becomes precise.

Error Bound Analysis

Error bound analysis is an essential aspect of integral estimation techniques because it provides information about the possible error involved with the numerical integration. For Simpson’s Rule, the error bound helps us understand how accurate the approximation is.
The error bound for Simpson's Rule can be formulated as:

• \( E_S = \frac{-(b-a)^5}{180n^4} \cdot M \)

where \( M \) is the maximum value of the absolute fourth derivative of the function in the interval \([a, b]\).
Traditionally, the smaller the fourth derivative's absolute maximum, the more accurate Simpson’s Rule will be, since this indicates that the function hasn't deviated much in curvature.
The practical application of Simpson's Rule, as demonstrated in the original exercise, is that when the fourth derivative is zero, the error bound becomes zero, implying a near-perfect approximation.
This emphasizes why understanding and calculating error bounds is crucial in determining the reliability of mathematical approximations.