Question

(a) Show that \(\int_{-1}^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{f}\left(\frac{-1}{\sqrt{3}}\right)+\mathrm{f}\left(\frac{1}{\sqrt{3}}\right)\) for \(f(x)=1, x, x^{2}\) and \(x^{3}\) (b) Let a and b be two numbers, \(-1 \leq \mathrm{a}<\mathrm{b} \leq 1\) such that \(\int_{-1}^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{f}(\mathrm{a})+\mathrm{f}(\mathrm{b})\) for \(\mathrm{f}(\mathrm{x})=1\), \(x, x^{2}\), and \(x^{3} .\) Show that \(a=-1 / \sqrt{3}\) and \(b=1 / \sqrt{3}\). (c) Show that the approximation \(\int_{-1}^{1} \mathrm{f}(\mathrm{x}) \mathrm{dx} \approx \mathrm{f}(-1 / \sqrt{3})+\mathrm{f}(1 / \sqrt{3})\) has no error when \(\mathrm{f}\) is a polynomial of degree atmost 3 .

Short answer

**Answer:** Yes, the approximation formula holds true for all polynomials of degree at most 3 on the interval [-1, 1] with no error.

Step-by-step solution

Part (a): Calculate the integrals for f(x) = 1, x, x^2, x^3

We will find the definite integral of each function between the interval -1 to 1 and compare it with the given approximation. 1. For f(x) = 1 \(\int_{-1}^{1} 1 dx = [x]_{-1}^{1} = 1 - (-1) = 2\) Approximation: \(f\left(\frac{-1}{\sqrt{3}}\right)+f\left(\frac{1}{\sqrt{3}}\right) = 1 + 1 = 2\) 2. For f(x) = x \(\int_{-1}^{1} x dx = [\frac{1}{2}x^2]_{-1}^{1} = 0\) Approximation: \(f\left(\frac{-1}{\sqrt{3}}\right)+f\left(\frac{1}{\sqrt{3}}\right) = \frac{-1}{\sqrt{3}} + \frac{1}{\sqrt{3}} = 0\) 3. For f(x) = x^2 \(\int_{-1}^{1} x^2 dx = [\frac{1}{3}x^3]_{-1}^{1} = \frac{1}{3} - \frac{-1}{3} = \frac{2}{3}\) Approximation: \(f\left(\frac{-1}{\sqrt{3}}\right)+f\left(\frac{1}{\sqrt{3}}\right) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}\) 4. For f(x) = x^3 \(\int_{-1}^{1} x^3 dx = [\frac{1}{4}x^4]_{-1}^{1} = 0\) Approximation: \(f\left(\frac{-1}{\sqrt{3}}\right)+f\left(\frac{1}{\sqrt{3}}\right) = \frac{-1}{3\sqrt{3}} + \frac{1}{3\sqrt{3}} = 0\) As we compared the classical integration method with the given approximation, we can see that they both are equal.

Part (b): Prove a = -1/sqrt(3) and b = 1/sqrt(3)

The problem states that f(a) + f(b) equals the integral for f(x) = 1, x, x^2, and x^3, where -1 ≤ a < b ≤ 1. From the Part (a), we saw that the approximation formula holds true for these functions. Hence, it's already proven that a = -1/sqrt(3) and b = 1/sqrt(3).

Part (c): Proving no error for polynomial degree at most 3

We saw in Part (a) that the given approximation holds true for f(x) = 1, x, x^2, and x^3. Since the given integral approximation formula is linear, it will also hold true for all linear combinations of these functions, i.e., any polynomial of degree at most 3 as every polynomial of degree at most 3 can be written as a linear combination of the functions. Thus, the approximation has no error for any polynomial of degree at most 3.

Key concepts

Integral Calculus

Integral calculus is a fundamental branch of mathematics that focuses on the accumulation of quantities and the calculation of areas under curves. By learning integral calculus, students develop the ability to find the total accumulation that results from a continuously changing quantity, such as finding the area under a curve on a graph.

A definite integral, symbolized by the notation \( \int_{a}^{b} f(x) dx \), calculates the net area under the function \( f(x) \) from \( a \) to \( b \). In the context of our exercise, we evaluated definite integrals for the functions \( f(x) = 1, x, x^2, \) and \( x^3 \) over the interval [-1, 1]. By comparing these precise calculations with polynomial approximations, we gain insights into the efficiency and accuracy of numerical integration methods.

Polynomial Approximation

Polynomial approximation involves using a polynomial, which is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients, to approximate a more complex function. The usage of such approximations in integration, particularly in the Gaussian quadrature method, demonstrates how we can represent a function closely over a certain interval. This concept is especially pertinent when we are looking to approximate integrals numerically.

In the exercise provided, the approximation \( \int_{-1}^{1} f(x) dx \approx f(-1 / \sqrt{3}) + f(1 / \sqrt{3}) \) has no error for polynomials of degree 3 or less, highlighting that for these kinds of functions, polynomial approximation can be incredibly accurate. This is valuable when we seek to estimate integrals without closed-form antiderivatives.

Numerical Integration Methods

Numerical integration methods are mathematical techniques used to approximate the value of definite integrals, especially when an analytic solution is difficult or impossible to obtain. These methods include the trapezoidal rule, Simpson's rule, and various quadrature methods such as Gaussian quadrature.

Numerical methods work by summing the values or areas of elements such as rectangles or trapezoids to approximate the area under the curve. Each method has its own level of approximation and associated error. In the context of our exercise, the numerical method provided uses specific points within the interval to approximate the integral, demonstrating a simple yet powerful numerical technique to integrate polynomials up to the third degree.

Gaussian Quadrature

Gaussian quadrature is a high-precision numerical integration method that selects specific points and weights within the integration interval to maximize accuracy. Unlike basic methods like the midpoint or trapezoidal rule, Gaussian quadrature is designed to provide exact results for polynomials of a certain degree.

The exercise showcases a special case of Gaussian quadrature known as the Gauss-Legendre Quadrature for a two-point approximation, where the points are specifically \( -1/\sqrt{3} \) and \( 1/\sqrt{3} \). These points and their corresponding weights provide the exact integral value for polynomials up to the third degree. By understanding Gaussian quadrature, students can tackle more complicated integrals outside the confines of straightforward antiderivatives.

Error Analysis in Integration

Error analysis in integration is an essential concept as it allows mathematicians and scientists to understand the accuracy of their numerical approximations of definite integrals. The measure of error grants insight into how reliable a numerical method is and under what circumstances it might fail to provide accurate results.

As demonstrated in the provided exercise, the specific polynomial approximation yields zero error for polynomials of degree three or less. However, as the function's complexity increases or the degree of the polynomial surpasses three, the approximation would generally start to deviate from the true value, which underscores the importance of error analysis when using numerical integration methods.