Question

Exact Simpson's Rule a. Use Simpson's Rule to approximate \(\int_{0}^{4} x^{3} d x\) using two subintervals \((n=2) ;\) compare the approximation to the value of the integral. b. Use Simpson's Rule to approximate \(\int_{0}^{4} x^{3} d x\) using four subintervals \((n=4) ;\) compare the approximation to the value of the integral. c. Use the error bound associated with Simpson's Rule given in Theorem 8.1 to explain why the approximations in parts (a) and (b) give the exact value of the integral. d. Use Theorem 8.1 to explain why a Simpson's Rule approximation using any (even) number of subintervals gives the exact value of \(\int_{a}^{b} f(x) d x,\) where \(f(x)\) is a polynomial of degree 3 or less.

Short answer

Answer: Simpson's Rule provides the exact value of the integral for any polynomial function of degree 3 or less with any even number of subintervals because their fourth (and higher) derivatives are zero. This fact means that the error bound associated with Simpson's Rule becomes zero, leading to an exact value for the integral.

Step-by-step solution

Understanding Simpson's Rule

Simpson's Rule is a numerical integration technique used to approximate definite integrals. The rule states that for a function \(f(x)\) that is twice differentiable on \([a, b]\) and n is an even number of subintervals, the integral can be approximated by the following formula: \[ \int_{a}^{b} f(x) dx \approx \frac{(b-a)}{3n} \left[ f(a) + 4\sum_{i=1}^{n/2} f(x_{2i-1}) + 2\sum_{i=1}^{(n/2)-1} f(x_{2i}) + f(b) \right] \] We need to approximate the integral \(\int_{0}^{4} x^3 dx\) using this formula.

Computing the exact value of the integral

In order to find the exact value of the integral, we need to first calculate the antiderivative of the function \(f(x) = x^3\). By the power rule, the antiderivative of \(f(x)\) is \(F(x)=\frac{1}{4}x^4 + C\). Now we can find the exact value of the integral as: \[ \int_{0}^{4} x^3 dx = F(4) - F(0) = \frac{1}{4}(4^4) = 64 \]

Using Simpson's Rule for n=2 subintervals

First, let's evaluate the approximation using two subintervals \((n=2)\): 1. Divide the interval \([0,4]\) into two equal subintervals: \(x_{0}=0\), \(x_{1}=2\), and \(x_{2}=4\). 2. Apply Simpson's Rule formula: \[ \int_{0}^{4} x^3 dx \approx \frac{(4-0)}{3\times2} \left[ f(0) + 4f(2) + f(4) \right] = \frac{8}{3} \left[ 0 + 4(8) + 64 \right] = 64 \] Comparing the approximation with the exact value, we can see that they are the same.

Using Simpson's Rule for n=4 subintervals

Now, let's evaluate the approximation using four subintervals \((n=4)\): 1. Divide the interval \([0,4]\) into four equal subintervals: \(x_{0}=0\), \(x_{1}=1\), \(x_{2}=2\), \(x_{3}=3\), and \(x_{4}=4\). 2. Apply Simpson's Rule formula: \[ \int_{0}^{4} x^3 dx \approx \frac{(4-0)}{3\times4} \left[ f(0) + 4f(1) + 2f(2) + 4f(3) + f(4) \right] \\ = \frac{4}{3} \left[ 0 + 4(1) + 2(8) + 4(27) + 64 \right] = 64 \] Comparing the approximation with the exact value, we can see that they are the same.

Using Theorem 8.1 to explain why the approximations are exact

Theorem 8.1 states that the error bound associated with Simpson's Rule is given by: \[ \left|\int_{a}^{b} f(x) dx - S_n\right| \leq \frac{(b-a)^5}{180n^4}K \] Where \(K\) is the value such that \(|f^{(4)}(x)| \leq K\) for all x in the interval \([a,b]\). For our case, \(f(x) = x^3\), so its fourth derivative is \(f^{(4)}(x) = 0\). So the error bound becomes zero. Hence, the approximations in parts (a) and (b) give the exact value of the integral.

Generalizing for any even number of subintervals

As we have shown in Step 5, the errors associated with Simpson's Rule approximations are zero for the integral of \(f(x) = x^3\), because its fourth derivative is zero. This fact can be generalized to any polynomial function of degree 3 or less, as their fourth (and higher) derivatives are also zero. Therefore, for any even number of subintervals, Simpson's Rule will provide the exact value of the integral.

Key concepts

Numerical Integration

Numerical integration is a cornerstone of computational mathematics, providing a method to approximate the area under a curve, which corresponds to the value of a definite integral. Rather than finding an antiderivative, which can be extremely challenging or sometimes impossible for complex functions, numerical integration uses geometric or algebraic techniques to estimate the integral's value.

One of the most prominent methods of numerical integration is Simpson's Rule. It's particularly useful as it strikes a balance between simplicity and accuracy. This rule approximates the area under a curve by dividing it into segments and fitting a second-degree polynomial (a parabola) through three consecutive points on the curve. The result is a series of 'curved trapezoids' that closely mirror the function's shape. The sum of the areas of these segments gives us an approximation of the integral.

Definite Integrals Approximation

Definite integrals are the foundation of calculus, representing the net area bounded by the graph of a function and the x-axis over a given interval. The process of finding this area exactly is called integration. However, in many practical scenarios, an exact solution is either not feasible or is excessively time-consuming, which is where approximation methods become invaluable.

Approximation of definite integrals through numerical methods can be likened to an artist's sketch of a landscape; it is not perfect, but it can often give a remarkably good representation of the scene. Simpson's Rule is one such method used to approximate definite integrals—especially when dealing with functions that are difficult to integrate analytically. The effectiveness of Simpson's Rule is largely due to its use of a sequence of parabolic arcs instead of straight lines to estimate the curve of the function between points.

Polynomial Function Integration

Integration of polynomial functions involves finding an antiderivative, a reverse operation to taking the derivative. For polynomials, the antiderivative can be found exactly using the fundamental theorem of calculus, which leads to precise calculations of areas under curves described by polynomial equations.

However, when applying Simpson's Rule, something interesting occurs. Since Simpson's Rule is based on parabolic interpolation, it can perfectly approximate integrals of polynomial functions of degree 3 or less. This is because the approximation segments themselves are parabolic, matching the exact shape of the polynomial's graph over each interval. Hence, when you apply Simpson's Rule to functions like the cubic polynomial in the exercise \(x^3\), you get the exact area without any approximation error. This characteristic highlights why Simpson's Rule is particularly advantageous for integrating polynomials up to the third degree.

Error Bound for Integration

One of the most vital considerations when approximating an integral is understanding the potential error in the approximation. Assessing this error allows for the determination of how accurate an approximation method is. The error bound for Simpson's Rule quantifies the maximum possible error between the true value of the integral and its approximation.

The error bound for Simpson’s Rule is theoretically determined and expressed in terms of the function's fourth derivative. It is given by \( \frac{(b-a)^5}{180n^4}K \), where \(K\) is a bound on the absolute value of the fourth derivative within the interval of integration. The smaller the value of \(K\), the tighter our error bound will be. In the exercise, the fourth derivative of the cubic polynomial is zero, meaning that the error bound reduces to zero, signifying that Simpson's Rule provides the exact integral value for the polynomial. This principle demonstrates the accuracy and reliability of Simpson's Rule for certain classes of functions.