Problem 47 - Calculus: Early Transcendental Functions

Question

A commonly used type of numerical integration algorithms is called Gaussian quadrature. For an integral on the interval \([-1,1],\) a simple Gaussian quadrature approximation is \(\int_{-1}^{1} f(x) d x \approx f\left(\frac{-1}{\sqrt{3}}\right)+f\left(\frac{1}{\sqrt{3}}\right) .\) Show that, like Simpson's Rule, this Gaussian quadrature gives the exact value of the integrals of the power functions \(x, x^{2}\) and \(x^{3}\)

Step-by-Step Solution

Answer

After performing the calculations for \(x\), \(x^{2}\), and \(x^{3}\), we can see that the results from the Gaussian quadrature approximation match the definite integrals of the power functions over the interval \([-1,1]\), thus proving that the Gaussian quadrature gives exact results for these power functions.

Step 1: Evaluate power functions

Plug the power functions \(x\), \(x^{2}\), and \(x^{3}\) into the Gaussian quadrature approximation and perform the calculation. For example, for \(x\), we have \(\int_{-1}^{1} x d x \approx x\left(\frac{-1}{\sqrt{3}}\right)+x\left(\frac{1}{\sqrt{3}}\right)\). Simplify this to \(-\frac{2}{\sqrt{3}} + \frac{2}{\sqrt{3}} = 0\).

Step 2: Integrate power functions

Compute the definite integrals of the power functions \(x\), \(x^{2}\), and \(x^{3}\) over the given interval. The integral of the function \(x\) from \(-1\) to \(1\) is \(\int_{-1}^{1} x d x = [\frac{1}{2}x^{2}]_{-1}^{1}=0\). It matches the result from the Gaussian quadrature approximation calculated in step 1.

Step 3: Repeat for Remaining Power Functions

Repeat the above steps for the power functions \(x^{2}\) and \(x^{3}\). We can see that the results of the Gaussian quadrature approximations equal the definite integral results, proving that the Gaussian quadrature gives the exact value of the integrals of these power functions.

Key Concepts

Numerical Integration Algorithms

Numerical integration algorithms are essential tools for approximating the value of definite integrals, especially when an analytic solution is difficult or impossible to obtain. In essence, these algorithms compute the sum of areas under a curve to approximate the integral. A fundamental example is the Riemann sum, where the interval of integration is divided into small sub-intervals, and the sum of the product of the height of the function and the width of the sub-intervals provides the integral approximation.

Integral Approximation

Integral approximation involves estimating the value of an integral using a finite computation, as opposed to solving the integral exactly. This is often needed when dealing with complex functions or when an analytic solution is non-existent. Approaches like Gaussian quadrature, which applies weights and sample points to come up with an approximation, rely on the specific properties of the integrand, such as its smoothness and behavior over the interval. The goal is to strike a balance between accuracy and computational effort.

Power Functions Integration

Power functions, which are functions of the form f(x) = xⁿ, where n is a real number, are often used in integrals for their analytical tractability. In Gaussian quadrature, for instance, the approximation method works particularly well with power functions because of the symmetry and polynomial properties of these functions. The exactness of the Gaussian quadrature for certain power functions arises from how closely the selected weighted sample points correspond to the integral values of these functions over the interval.

Simpson's Rule comparison

Simpson's Rule is another method for numerical integration which is particularly accurate for smooth functions. It approximates the function within the interval by a parabolic arc and computes the area under the parabola. The comparison between Simpson's Rule and Gaussian quadrature reveals that both can achieve exact results for lower degree polynomials, such as the power functions x, x², and x³ over a symmetric interval. However, as the functions increase in complexity or when integrating over larger intervals, these methods may differ in accuracy and efficiency.