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Chapter 4: Problem 1

Prove that the sum of the weights in Newton-Cotes rules is \(b-a,\) for any \(n\).

Short Answer

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Based on the given step-by-step solution, the short answer to this question is as follows: To prove that the sum of weights in Newton-Cotes rules is b-a for any n, we need to calculate the weights for a generic function using the method of undetermined coefficients. By considering polynomials up to degree n and applying the method, we equate the left-hand and right-hand sides of the equations until we obtain the simplified expression (b - a) = 鈭�(i = 0 to n) W_i. This expression demonstrates that the sum of the weights in Newton-Cotes rules is indeed equal to b-a for any n.

Step by step solution

01

Understand Newton-Cotes rules and their weights

Newton-Cotes rules are numerical integration techniques used to approximate the definite integral of a function over a closed interval [a, b] using n+1 equally spaced points. These techniques are based on the idea of fitting a polynomial to the function and integrating the polynomial over the interval. The weights in Newton-Cotes rules are the coefficients that are multiplied by the function values at each point to approximate the definite integral.
02

Calculate the weights for a generic function

Let's calculate the weights for a generic function f(x) being integrated on the interval [a, b] using n+1 equally spaced points. The spacing between each point will be h = \frac{b-a}{n}. The Newton-Cotes formula can be written as: \int_{a}^{b}f(x)dx 鈮� \sum_{i=0}^{n} W_i f(x_i) Where W_i is the weight at the i-th point, and x_i = a + ih for i = 0, 1, 2, ..., n.
03

Use the method of undetermined coefficients

To calculate the weights W_i, we can use the method of undetermined coefficients. To do this, we consider polynomials up to the (n+1)th degree and find the weights for which the approximation is exact. Let g(x) be a polynomial of degree n or less. Then, by the method of undetermined coefficients, we can find the weights W_i that make the following equation true: \int_{a}^{b} g(x) dx = \sum_{i=0}^{n} W_i g(x_i)
04

Calculate the integral of g(x)

Before we can solve for the weights, we need to calculate the left-hand side of the previous equation, which is the integral of g(x) on the interval [a, b]: \int_{a}^{b} g(x) dx = \int_{a}^{b} \sum_{j=0}^{n} g_j x^j dx Where g_j are the coefficients of the polynomial g(x). By integrating term by term, we can rewrite the integral as: \int_{a}^{b} g(x) dx = \sum_{j=0}^{n} g_j \int_{a}^{b} x^j dx Now, let's find the integral of x^j on the interval [a, b]: \int_{a}^{b} x^j dx = \frac{1}{j+1} (b^{j+1} - a^{j+1})
05

Apply the method of undetermined coefficients

Applying the method of undetermined coefficients, we equate the left-hand side and the right-hand side of the equations derived above: \sum_{j=0}^{n} g_j \int_{a}^{b} x^j dx = \sum_{i=0}^{n} W_i \sum_{j=0}^{n} g_j x_i^j Which can be simplified to: \sum_{j=0}^{n} g_j \frac{1}{j+1} (b^{j+1} - a^{j+1}) = \sum_{i=0}^{n} W_i \sum_{j=0}^{n} g_j x_i^j
06

Prove the sum of the weights equals b-a

Now, let's consider the j=0 special case of the equation: \sum_{j=0}^{0} g_j \frac{1}{j+1} (b^{j+1} - a^{j+1}) = \sum_{i=0}^{n} W_i \sum_{j=0}^{0} g_j x_i^j This simplifies to: g_0 \cdot \frac{1}{1} (b - a) = \sum_{i=0}^{n} W_i \cdot g_0 Since g_0 can be factored out, we have: (b - a) = \sum_{i=0}^{n} W_i This proves that the sum of the weights in Newton-Cotes rules is b-a, for any n.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Newton-Cotes Rules
Newton-Cotes rules are essential for those tackling numerical integration problems. These rules provide a way to approximate the definite integral of a function over a set interval, [\(a, b\)], using a polynomial.
  • The interval is divided into \(n+1\) equally spaced points.
  • Weights are assigned to these points to approximate the area under the curve.
Newton-Cotes rules come in various forms like the Trapezoidal Rule and Simpson's Rule. These rules undergo polynomial approximation to fit the curve of the target function. The weights in the Newton-Cotes formulas ensure these approximations sum to particular values, like the length of the interval \(b-a\). Different degrees of polynomials yield different versions of Newton-Cotes rules, each with its benefits regarding accuracy and complexity.
Method of Undetermined Coefficients
The method of undetermined coefficients is a powerful tool used to derive the specific weights for integrals in Newton-Cotes rules. This method involves:
  • Assuming a general polynomial function \(g(x)\).
  • Using the polynomial to find weights that make the Newton-Cotes formula exact.
By setting up equations where the polynomial's integral over the interval equals the approximate sum of the function values at chosen points, we solve for these weights. This process is known as fitting or approximating the function exactly using undetermined coefficients.The beauty of this method lies in its ability to provide an exact solution for specific cases, particularly when the function being integrated can be closely approximated by polynomial terms.
Polynomial Approximation
Polynomial approximation serves as the backbone of many numerical methods, including Newton-Cotes rules. It involves using a polynomial to mimic the behavior of a complex function to solve integrals more easily.
  • A polynomial can efficiently approximate many types of functions over small intervals.
  • The degree of the polynomial often determines the approximation's accuracy.
By approximating the function with polynomials up to degree \(n\), numerical integration becomes a manageable task. The choice of degree directly affects the Newton-Cotes rule's precision and is crucial for achieving the desired balance between simplicity and accuracy.Polynomial approximation makes it feasible to perform integrations that otherwise might be too complex or computationally expensive.
Definite Integrals
Definite integrals calculate the signed area under a curve between two points \(a\) and \(b\) on the x-axis. In numerical integration, we focus on approximating this area when the function is too complex to integrate analytically.
  • Numerical methods like Newton-Cotes rules are used to find these integrals.
  • The function range \([a, b]\) is divided into intervals, making complex integrations more straightforward.
Definite integrals are foundational in both mathematics and physics, representing quantities like displacement and area. By converting these integrals into manageable calculations, numerical integration helps solve real-world problems effectively and accurately.

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Most popular questions from this chapter

Prove that the sum of the weights in Gauss-Legendre quadrature is \(2,\) for any \(n\).

Approximate \(\int_{1}^{1.5} x^{2} \log x d x\) using Gauss-Legendre rule with \(n=2\) and \(n=3\). Compare the approximations to the exact value of the integral.

Determine the value of \(n\) required to approximate $$ \int_{0}^{2} \frac{1}{x+1} d x $$ to within \(10^{-4}\), and compute the approximation, using the composite trapezoidal and composite Simpson's rule.

Show that the absolute error for the composite trapezoidal rule decays at the rate of \(1 / n^{2},\) and the absolute error for the composite Simpson's rule decays at the rate of \(1 / n^{4}\), where \(n\) is the number of sub intervals.

A composite Gauss-Legendre rule can be obtained similar to composite Newton- Cotes formulas. Consider \(\int_{0}^{2} e^{x} d x .\) Divide the interval (0,2) into two sub intervals (0,1),(1,2) and apply the Gauss-Legendre rule with two- nodes to each sub interval. Compare the estimate with the estimate obtained from the Gauss-Legendre rule with four-nodes applied to the whole interval (0,2)
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