Question

Show in detail how to obtain the Simpson rule $$ \int_{a}^{b} f(x) \mathrm{d} \mathrm{x} \approx \frac{b-a}{6}\left[f(a)+4 f\left(\frac{a+b}{2}\right)+f(b)\right] $$

Short answer

Answer: The Simpson rule is a numerical integration technique that approximates the definite integral of a function by fitting a quadratic curve to three data points. It is derived using polynomial interpolation in the following steps: 1. Perform polynomial interpolation for the given function using a quadratic polynomial P(x) that passes through three points (a, f(a)), (m, f(m)), and (b, f(b)), where m = (a+b)/2. 2. Set up a system of equations involving A, B, and C for the quadratic polynomial P(x) = Ax^2 + Bx + C. 3. Solve the system of equations to find the values of A, B, and C. 4. Approximate the definite integral of f(x) over the interval [a, b] by calculating the integral of P(x) over the same interval. 5. Derive the Simpson rule by computing the individual integrals in step 4, which results in the formula: $$\int_{a}^{b} f(x) \mathrm{d} \mathrm{x} \approx \frac{b-a}{6}\left[f(a)+4 f\left(\frac{a+b}{2}\right)+f(b)\right]$$

Step-by-step solution

Polynomial interpolation of the given function

Given a function, f(x), polynomial interpolation involves determining a polynomial that approximates the function well. In the case of the Simpson rule, we will use quadratic interpolation, where we determine a quadratic polynomial P(x) that passes through three points (a, f(a)), (m, f(m)), and (b, f(b)), where m = (a+b)/2.

Quadratic polynomial in terms of a, b, and m

Let \(P(x) = Ax^2 + Bx + C\). Now, we set up three equations based on the fact that P(x) should pass through the points (a, f(a)), (m, f(m)), and (b, f(b)) as shown below: 1. \(Aa^2 + Ba + C = f(a)\) 2. \(Am^2 + Bm + C = f(m)\) 3. \(Ab^2 + Bb + C = f(b)\)

Solving for A, B, and C

Next, we need to solve the system of equations created in Step 2 to get A, B, and C. As m = (a+b)/2, we plug this value into equation 2 and simplify the result. Subtracting the first equation from the third, we get: \(B(b-a) = f(b) - f(a) - A(b^2 - a^2)\) We know that \(b^2 - a^2 = (b-a)(b+a)\). So, we can rewrite the equation above as: \(B = \frac{f(b) - f(a) - A(b-a)(b+a)}{b-a}\) Plug this value of B into the expression we found in equation 2 after replacing m: \(A\left(\frac{a+b}{2}\right)^2 + \frac{f(b) - f(a) - A(b-a)(b+a)}{b-a}*\frac{a+b}{2} + C = f\left(\frac{a+b}{2}\right)\) By solving for A and C, and substituting these values into the original expression of P(x), we arrive at the desired quadratic polynomial.

Approximating the definite integral

Now that we have a quadratic polynomial \(P(x) = Ax^2 + Bx + C\), we can approximate the definite integral of f(x) over the interval [a, b] by calculating the integral of P(x) over the same interval: $$\int_{a}^{b} f(x) \mathrm{d} \mathrm{x} \approx \int_{a}^{b} P(x) \mathrm{d} \mathrm{x}$$

Deriving the Simpson rule

To derive the Simpson rule, calculate the integral of P(x) found in step 3: $$\int_{a}^{b} P(x) \mathrm{d} \mathrm{x} = \int_{a}^{b} (Ax^2 + Bx + C) \mathrm{d} \mathrm{x} = A\int_{a}^{b} x^2 \mathrm{d} \mathrm{x} + B\int_{a}^{b} x \mathrm{d} \mathrm{x} + C\int_{a}^{b} \mathrm{d} \mathrm{x}$$ After computing these integrals individually, we end up with the Simpson rule: $$\int_{a}^{b} f(x) \mathrm{d} \mathrm{x} \approx \frac{b-a}{6}\left[f(a)+4 f\left(\frac{a+b}{2}\right)+f(b)\right]$$

Key concepts

Polynomial Interpolation

Polynomial interpolation is a process used to construct a polynomial that passes through a set of known points. This is particularly useful when trying to approximate a complex function that might be difficult to integrate analytically. By substituting the original function with the interpolated polynomial, we simplify complex calculations.

Application of Polynomial Interpolation in Numerical Methods

In the context of numerical integration methods, such as Simpson's rule, polynomial interpolation serves as the foundation for approximate calculations. The premise is to replace the complex function with an easier-to-integrate polynomial that approximates it well over a certain interval. The more accurate the interpolation, the closer the approximation will be to the actual value of the integral.

Definite Integral Approximation

The definite integral of a function provides the net area under the curve between two points. However, not all functions can be integrated exactly using elementary functions. In such cases, we turn to numerical methods that allow us to estimate the value of the integral.

Utility of Definite Integral Approximation

Approximation techniques come in handy when the function's antiderivative is unknown or when a quick estimation is needed. For students and engineers alike, being able to approximate the integral means they can still solve problems and make predictions even when an exact solution is unattainable.

Numerical Integration Methods

Numerical integration is a part of numerical analysis focusing on the algorithms for approximating the value of a definite integral. Simpson's rule, which this exercise demonstrates, is one such method. It's used when an exact integral of a function is difficult to find or when dealing with data points rather than functions.

Variety of Numerical Integration Techniques

There are several methods available, such as the trapezoidal rule, Simpson’s rule, and Gaussian quadrature. Each technique involves different levels of complexity and accuracy, providing a robust toolkit for tackling various integral approximation challenges.

Quadratic Polynomial Interpolation

Quadratic polynomial interpolation involves fitting a second-degree polynomial to a set of data points. It is a specific case of polynomial interpolation that yields a quadratic function, which is a parabola. For Simpson’s rule, we use three data points to determine a quadratic polynomial that closely matches the portion of the original function over a small interval.

How Quadratic Interpolation Aids in Integration

The advantage of using a quadratic polynomial lies in its perfect balance between simplicity and the ability to approximate a curve. Since it's a straightforward parabolic shape, it's often easier to integrate than higher-degree polynomials or complex functions, while still being flexible enough to provide a good estimation over small intervals.