Numerical integration: Most numerical methods for evaluating...

Numerical integration: Most numerical methods for evaluating a definite integral $\int_a^b f(x) d x$ are based on interpolation. One chooses $n+1$ interpolation points $a \leq x_0<$ $x_1<\cdots<x_n \leq b$ and replaces the integrand by its interpolating polynomial $p_n(x)$ of degree $n$, leading to the approximation $\int_a^b f(x) d x \approx \int_a^b p_n(x) d x$, where the polynomial integral can be done explicitly. Write down the following popular integration rules: (a) Trapezoid Rule: $x_0=a, x_1=b$. (b) Simpson's Rule: $x_0=a, x_1=\frac{1}{2}(a+b), x_2=b$. (c) Simpson's $\frac{3}{8}$ Rule: $x_0=a, x_1=\frac{1}{3}(a+b), x_2=\frac{2}{3}(a+b), x_3=b$. (d) Midpoint Rule: $x_0=\frac{1}{2}(a+b)$. (e) Open Rule: $x_0=\frac{1}{3}(a+b), x_1=\frac{2}{3}(a+b)$. (f) Test your methods for accuracy on the following integrals: (i) $\int_0^1 e^x d x$ (ii) $\int_0^\pi \sin x d x$, (iii) $\int_1^e \log x d x$ (iv) $\int_0^{\pi / 2} \sqrt{x^3+1} d x$. Note: For more details on numerical differentiation and integration, you are encouraged to consult a basic numerical analysis text, e.g., [8].

Numerical integration: Most numerical methods for evaluating a definite integral $\int_a^b f(x) d x$ are based on interpolation. One chooses $n+1$ interpolation points $a \leq x_0<$ $x_1<\cdots<x_n \leq b$ and replaces the integrand by its interpolating polynomial $p_n(x)$ of degree $n$, leading to the approximation $\int_a^b f(x) d x \approx \int_a^b p_n(x) d x$, where the polynomial integral can be done explicitly. Write down the following popular integration rules:
(a) Trapezoid Rule: $x_0=a, x_1=b$. (b) Simpson's Rule: $x_0=a, x_1=\frac{1}{2}(a+b), x_2=b$.
(c) Simpson's $\frac{3}{8}$ Rule: $x_0=a, x_1=\frac{1}{3}(a+b), x_2=\frac{2}{3}(a+b), x_3=b$.
(d) Midpoint Rule: $x_0=\frac{1}{2}(a+b)$. (e) Open Rule: $x_0=\frac{1}{3}(a+b), x_1=\frac{2}{3}(a+b)$.
(f) Test your methods for accuracy on the following integrals:
(i) $\int_0^1 e^x d x$
(ii) $\int_0^\pi \sin x d x$,
(iii) $\int_1^e \log x d x$
(iv) $\int_0^{\pi / 2} \sqrt{x^3+1} d x$.
Note: For more details on numerical differentiation and integration, you are encouraged to consult a basic numerical analysis text, e.g., [8].

Key Concepts

Open Rule

Open integration rules, unlike the closed rules where the endpoints are used, generate approximations based on interior points only. These rules are beneficial when the integrand is undefined or exhibits singular behavior at the endpoints. By avoiding endpoint evaluations, open rules can offer better stability or accuracy in such cases.

Midpoint Rule

The Midpoint Rule approximates the definite integral by evaluating the integrand at the midpoint of the interval and multiplying that value by the length of the interval. This method effectively approximates the function by a constant, representing its average value over the interval. It is particularly effective when the function is relatively constant or when a rough estimate of the area is sufficient.

Simpson's 3/8 Rule

Simpson's 3/8 Rule is similar to Simpson's Rule but uses a cubic polynomial approximation over four points. It is particularly useful for integrating functions over subintervals where a higher degree polynomial provides a more accurate representation of the function's behavior. The weightings in the Simpson's 3/8 Rule reflect the integration of a cubic interpolant, providing an improved approximation when the function displays more curvature.

Simpson's Rule

Simpson's Rule is a numerical integration technique that approximates the integrand by a quadratic polynomial. By using three equally spaced points (the endpoints and the midpoint), it computes the weighted average of the function values such that the approximating polynomial passes through these points. Simpson's Rule generally provides higher accuracy than the Trapezoidal Rule for smooth functions, especially when the function resembles a quadratic behavior over the interval.

Polynomial Interpolation

Polynomial interpolation involves approximating a function by a polynomial that passes through a set of given points. When applied to numerical integration, the idea is to replace the integrand by its interpolating polynomial, which can be integrated exactly over the interval. The accuracy of the approximation depends on the number and distribution of the interpolation points and the degree of the polynomial.

Interpolatory Quadrature

Interpolatory quadrature rules are numerical integration techniques that estimate the integral by integrating an interpolating polynomial that approximates the function. These rules, including the Newton-Cotes formulas, use the values of the function at specific points in the integration interval to calculate weighted sums, where the weights are determined by the integration of the corresponding polynomial basis functions.

Numerical Integration

Numerical integration refers to a collection of algorithms for approximating the value of definite integrals. Instead of finding an antiderivative analytically, numerical methods estimate the area under a curve using a finite set of function evaluations. This is particularly useful when the integrand is complex, does not have an elementary antiderivative, or when only discrete data points are available.

Trapezoidal Rule

The Trapezoidal Rule approximates the integral of a function by replacing the area under the curve with a single trapezoid. It uses the function values at the two endpoints of the interval, effectively approximating the integrand by a straight line. This method is simple and works well for functions that are close to linear over the interval, though its accuracy decreases for more curved functions.

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