Consider the following integrals and the given values of n ....

Consider the following integrals and the given values of $n .$ a. Find the Trapezoid Rule approximations to the integral using n anc 2n subintervals. b. Find the Simpson's Rule approximation to the integral using $2 n$ subintervals. It is easiest to obtain Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example 6 c. Compute the absolute errors in the Trapezoid Rule and Simpson's $$ \int_{1}^{e} \frac{1}{x} d x ; n=50 $$

Consider the following integrals and the given values of $n .$
a. Find the Trapezoid Rule approximations to the integral using n anc
2n subintervals.
b. Find the Simpson's Rule approximation to the integral using $2 n$ subintervals. It is easiest to obtain Simpson's Rule approximations
from the Trapezoid Rule approximations, as in Example 6
c. Compute the absolute errors in the Trapezoid Rule and Simpson's
$$
\int_{1}^{e} \frac{1}{x} d x ; n=50
$$

Key Concepts

Composite Numerical Integration

Composite Numerical Integration involves applying numerical integration rules, such as the Trapezoidal or Simpson's Rule, over several smaller subintervals rather than a single large interval. This strategy improves the approximation by reducing local errors and making it possible to handle more complex or non-linear functions, with adjustments like using different numbers of subintervals (e.g., n, 2n) to refine the result.

Error Analysis in Numerical Integration

Error Analysis in Numerical Integration assesses the difference between the true value of an integral and its numerical approximation. This analysis computes absolute or relative errors, which help in understanding the accuracy of a chosen numerical method. Evaluating the errors from different methods, like the Trapezoidal and Simpson's Rules, can indicate which provides a more precise approximation for a given number of subintervals.

Trapezoidal Rule

The Trapezoidal Rule is a numerical method for approximating definite integrals by dividing the interval of integration into subintervals and approximating the area under the curve on each subinterval with a trapezoid. This method computes the average of the function’s values at the endpoints of each subinterval and sums these areas, offering a simple approach that is particularly useful when an antiderivative is difficult to obtain analytically.

Simpson's Rule

Simpson's Rule is another technique for numerical integration that provides a more accurate approximation by using parabolic arcs rather than straight lines to approximate the function over subintervals. By fitting quadratic polynomials through sets of three adjacent points, Simpson’s Rule typically yields better accuracy than the Trapezoidal Rule, especially for functions that are smooth, and is based on a weighted average that combines evaluations at equally spaced points.

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